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Math 335 HandoutsSectionTopicPage NumberGraphing ReviewFunctions and Circles3-14Appendices A, B and CEquations, Inequalities and Functions151.1Angles171.2Angle Relationships and Similar Triangles213.1 (part I)Radian Measures23WorksheetWarm-up251.3Trig Functions271.4Using Definitions of Trig Functions312.1Trig Functions of Acute Angles352.2Trig Functions of Non-acute Angles393.1 (Part II)Radian Measures393.2Applications of Radian Measures413.3The Unit Circle and Circular Functions432.4Solving Right Triangles47WorksheetWord Problems Practice492.5Further Applications of Right Triangles517.1Oblique Triangles and The Law of Sines537.3The Law of Cosines597.2The Ambiguous Case of the Law of Sines63SummarySolving a Triangle684.1Sine and Cosine graphs694.2Translations of Sine and Cosine Graphs754.3Tangent and Cotangent Graphs794.4Secant and Cosecant Graphs834.5Harmonic Motion85worksheetTrig Graphing Review875.1Fundamental Identities935.2Verifying Trig Identities955.3Sum and Difference Identities for Cosine995.4Sum and Difference Identities for Sine and Tangent1035.5Double Angle Identities107WorksheetIdentities Review1115.6Half Angle Identities1156.1Inverse Circular Functions1196.2Trig Equations I127WorksheetFactoring Review1316.3Trig Equations II1336.4Equations Involving Inverse Trig Functions135worksheetSolving Trig Equations Practice137worksheetExponential and logarithmic functions1397.4Vectors, Operations and Dot Products1417.5Applications of Vectors (lecture examples)1477.5Applications of Vectors (exercises)1498.1Complex Numbers1518.2Polar Form of Complex Numbers1558.3The Product and Quotient Theorems1598.4The DeMoivre’s Theorem; Powers and Roots and Complex Numbers1618.5Polar Equations and Graphs165worksheetThe Cycle of a Rose Curve171worksheetPolar Equations and Graphs173HomeworkPolar Equations and Graphs177TemplatesHomework FormsBack of packetHW Form A (x4) - 16 copies HW Form B (x6) - 13 copies HW Form C (graph) – 9 copies HW Form D (graph) – 5 copiesHW Form E (x3) - 26 copies HW Form F (x2) – 6 copiesBelow is the list of all sections involving required problems and the type of HW form to be used with each. Start each new section on a new form. Work in boxes LEFT TO RIGHT!SectionHW FormSectionHW FormApp AA5.6E (x2)App BD6.1A (x2)App CA6.2E (x2)1.1B6.3E (x2)1.2B6.4A3.1 (Part I)Bexponential/logsworksheets1.3D & A8.1E1.4B7.4B (x2)2.1B7.5E2.2B8.2A3.1 (part II)A8.3E3.2B8.4E3.3B8.5E (x2)3.4A2.4A2.5E7.1E (x2)7.3E (x2)7.2E (x2)4.1C (x2)4.2A & C4.3C (x2)4.4C4.5F5.1A5.2F (x2)5.3A5.4E (x2)5.5E (x2)Key Functions: Absolute Value*Label three ordered pairs on the function’s graph on both sides of the vertex. (integer coordinates only)*Use interval notation to list the following:Domain:Range:I.Sketch each function on the given grid. List the coordinates for the vertex.a)b)c)d) II.Evaluate and simplify each expression.a)b)c)Determine if the following statement is true or false. Show your work. = ?III.Solve each equation for all values of ‘x’ given that .a)b) c)IV.Write the function for each graph.a)b) c)Key Functions: Quadratic*Label seven ordered pairs on the function’s graph. (integer coordinates only)*Use interval notation to list the following:Domain:Range: I.Sketch each parabola on the provide grids. List the coordinates of the vertex for each.a)b)c)d) II.Evaluate and simplify each expression given a)b)c)d)III.Solve each equation for all values of ‘x’ given that . a)b)IV.Write the function for each graph.a)b)c)Key Functions: Cubic*Label five ordered pairs on the function’s graph. (integer coordinates only)*Use interval notation to list the function’s:Domain:Range:I.Sketch each function on the given grid. List the coordinates of the critical point.a)b)c)d)II.Evaluate and simplify each expression given that .a)b)c)d)III.Solve each equation for all values of ‘x’ given that . a)b)IV.Write the function for each graph.a)b)c)Key Functions: Rational*Label six ordered pairs on the function’s graph. (3 in each quadrant.)Label the Asymptotes on the function’s graph.*Use interval notation to list the following:Domain:Range:I.Sketch the graph on the given grid. Sketch the asymptotes with dotted lines and label them.a)b)c)d)II.Evaluate and simplify each expression given that .a)b)c)III.Solve each equation for all values of ‘x’ given that .a)b)IV.Write the function for each graph.Draw and label the asymptotes.a)b)c)Key Functions: Square Root*Label four ordered pairs on the function’s graph. (integer coordinates only)*Use interval notation to list the following:Domain:Range:I.Sketch each function on the given grid. a)b)c)d)II.Evaluate and simplify each expression given that .a)b)c)III.Solve each equation for all values of ‘x’ given that .a)b)c)IV.Write the function for each graph.a)b)c)Circles_______________________Circle:the set of points in a plane that are equidistant from a given point.List the center and radius of each circle.1.center _________Radius: r = _______2.center _________Radius: r = _______3.center _________Radius: r = _______4.center _________Radius: r = _______5.center _________Radius: r = _______6. center _________Radius: r = _______7. center _________Radius: r = _______8.center _________Radius: r = _______Graph each circle.9.10. 11.12.Appendices A, B and C Class practiceName :_________________________Date :___________________1.Equations and InequalitiesSolve. Write your final answer using set notation or interval notation.(a) (b) (c) 2.Functions(a) A relation is a set of ordered pairs.(b) A FUNCTION is a relation where ____________________________________________________.(c) Domain of a function is _____________________. (d) Range of a function is_________________.(e) 8 key functions studied in Intermediate Algebra:Function NameGeneral EquationSketch of graphDomain and RangeLinearQuadraticAbsolute ValueCubicRadicalRationalExponentialLogarithm3.List the Domain of each function. (Use interval notation.)(a) (b) (c) 4.Circle all functions with a Range of all Real Numbers.5.List the Domain and Range for each graph. (Use interval notation.)D: _________________R: ____________ D: ______________ R:______________Problems 6 - 7:Use the list of functions to respond to each question.6.Evaluate each of the following. Simplify if possible.(a) (b) (c) (d) 7.Solve for all values of x.Chapter 1: Trigonometric FunctionsTrigonometry: The study of triangles.Sec 1.1: AnglesDefinitionsA line is an infinite set of points where between any two points, there is another point on the line that lies between them.Line AB, ABA line segment is part of a line that consists of two distinct points on the line and all the points between them.Line segment AB or segment AB, ABA ray AB, denoted AB. The endpoint of the ray AB is A. The endpoint of the ray BA is B.An angle is a plane figure formed by two rays that share a common point (vertex).13761220terminal sideinitial side0terminal sideinitial sideInitial and terminal sidesvertexDegree is a unit for measuring angles and arcs that corresponds to 1/360 of a complete revolution.Counterclockwise rotation results in (+) measureClockwise rotation results (-) measureTypes of AnglesMeasurementAcute angleRight angleObtuse angleStraight angleNote: Angles are often denoted by the greek letter (theta). Other greek letters to represent angles are etc…..DefnsComplementary angles are angles whose sum measures that add up to 90° (they’re complements of each other)Supplementary angles are angles whose sum measures that add up to 180° (they’re supplements of each other)Ex 1Find the measure of the marked angles.a = (4x + 12)°, b = (2x + 66)°Ex 2Find the measure of the smaller angle formed by the hands of a clock at the given time.a) 4:00b) 1:45What do we do when we have an angle that is less than 1 degree?Use FRACTIONS of DEGREES with minutes and seconds.-13906598425Portions of a degree are measured with minutes and secondsDefnsOne minute, written 1', is 1/60 of a degree. 1'=(1/60)° or 60'=1°One second, written 1'' is 1/60 of a minute. 1''=(1/60)'=(1/3600)° or 60''=1'0Portions of a degree are measured with minutes and secondsDefnsOne minute, written 1', is 1/60 of a degree. 1'=(1/60)° or 60'=1°One second, written 1'' is 1/60 of a minute. 1''=(1/60)'=(1/3600)° or 60''=1'Note: Fractions of degrees are used in distance between two cities measured by latitude and longitude. Also, a space shuttle traveling thousands of miles will be far off its target if its heading is off by minutes or even seconds of a degree.Ex 3 (#48)Perform the calculation.(a) + (b) 90°-36°18'47'' ______________Ex 4 (#60)Convert the angle measure to degrees. If applicable, round to the nearest thousandth of a degree.165°51'09''Ex 5 (#72)Convert the angle measure to degrees, minutes, and seconds. Round answer to the nearest second, if applicable.122.6853°731573152DefnsAn angle is in standard position if its vertex is at the origin and its initial side is on the x-axis.Angles in standard position whose terminal sides lie on the x- or y-axis, such as angles with 90°, 180°, 270°, and so on, are called quadrantal angles.Angles with the same initial and terminal sides, but different amounts of rotation are called coterminal angles. Their measures differ by a multiple of 360°.0DefnsAn angle is in standard position if its vertex is at the origin and its initial side is on the x-axis.Angles in standard position whose terminal sides lie on the x- or y-axis, such as angles with 90°, 180°, 270°, and so on, are called quadrantal angles.Angles with the same initial and terminal sides, but different amounts of rotation are called coterminal angles. Their measures differ by a multiple of 360°.Ex 6 (#100)Give an expression that generates all angles coterminal with the angle 225°. Let n represent any integer.Ex 7Find the angle with least positive measure coterminal with each of the following angles.(a) 260° (b) -750°.Ex 8 (#124)Locate the point -22, 22 and draw a ray from the origin to the point. Indicate with an arrow the angle in standard position having least positive measure. Then find the distance r from the origin to the point, using the distance formula.Ex 9 (#134)An airplane propeller rotates 1000 times per min. Find the number of degrees that a point on the edge of the propeller will rotate in 1 sec.Try Problems:Sec 1.2: Angle Relationships and Similar TrianglesSketch an angle. How to name it?Review 426402550800mn34217865mn34217865Vertical angles (have same measure)Parallel linesTransversalAlternate interior angles (equal) Alternate exterior angles (equal)Interior angles on the same side of transversal (suppl.)Corresponding angles (equal)50800119380FactThe sum of the measure of the angles of any triangle is 180°00FactThe sum of the measure of the angles of any triangle is 180°Review Types of TrianglesCharacteristicsAcute TriangleRight TriangleObtuse TriangleEquilateral TriangleIsosceles TriangleScalene Triangle-343535153670Conditions for Similar TrianglesFor any triangle ABC to be similar to triangle DEF, the following conditions must hold.Corresponding angles must have the same measure.Corresponding sides must be proportional. (That is, the ratios of their corresponding sides must be equal.)00Conditions for Similar TrianglesFor any triangle ABC to be similar to triangle DEF, the following conditions must hold.Corresponding angles must have the same measure.Corresponding sides must be proportional. (That is, the ratios of their corresponding sides must be equal.)Ex 1 Find the measures of angles 1, 2, 3, and 4 in the figure, given that the lines m and n are parallel.13335165735mn9x+9°12347x-5°mn9x+9°12347x-5°Ex 2 Find the measure of the marked angles.a = (11x + 12)°, b = (4x + 123)°Ex 3 Find side x with the given information.a = 13b = 12c = 5d = 26e = 24Ex 4A tree casts a shadow 42 m long. At the same time, the shadow cast by a 30-centimeter-tall statue is 66 cm long. Find the height of the tree. Include a sketch.Sec 3.1 (part I): Radian MeasureSo far, we have seen angles measured in degrees. Another unit of measurement for angles is radians.Radian measure is a common unit of measure. In more theoretical work in math, radian measure is preferred as it allows us to treat the domain of trig functions as real numbers, rather than angles. It also simplifies theorems such as the derivative of the sine function. If you plan on moving on to Calculus, you must be able to work in radians. It was wide applications in engineering and science.-11684078410DefnAn angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of 1 radian. radian think radiusDefnAn angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of 1 radian. radian think radiusEx1 Draw figures to represent θ=1/2 radian, θ=2 radians and θ=2π radians.Ex 2What quadrant is θ=5 radians in?-15545-216382Converting Between Degrees and Radians360°=2πDeg→RadMultiply by π/180°Rad→DegMultiply by 180°/πNoteIf no unit measure is specified, then the angle is understood to be measured in radians.Converting Between Degrees and Radians360°=2πDeg→RadMultiply by π/180°Rad→DegMultiply by 180°/πNoteIf no unit measure is specified, then the angle is understood to be measured in radians.Ex 3Convert each to degree or radian measure.a)b)c)d)108° -135° 11π12 -7π6Common Measures in Radians and Degrees469635810287Figure 4 on page 970Figure 4 on page 97Must know these equivalencesDegreesRadians (exact)Radians (approx.)0°030°π645°π460°π390°π2180°π270°3π228670255524500Try problems:Convert each to degree or radian measure.1) 2) 63) 4) Study these common angles before filling out the following chart without looking. Don’t peek!DegreesRadians (exact)0°30°45°60°90°180°270° Warm UpName:____________________________Date :______________Show complete solutions and circle all final answers. 1. Find the complement to an angle measuring .2. Find the angle of least positive measure that is coterminal with 3. Locate the point on a rectangular coordinate system.a) Draw a ray from the origin through the given point. b) Indicate with an arrow the least positive angle in standard position.c) Find the distance r from the origin to the point.4. Find the measure of each angle. (see diagram)5. The triangles are similar. Find ∠B. Note that and .6. Joey wants to know the height of a tree in a park near his home. The tree casts a 38-ft shadow at the same time that Joey, who is 63 inches tall, cast a 42-in shadow. Find the height of the tree.Sec 1.3: Trigonometric FunctionsWe will define the six trig functions: sine, cosine, tangent, cosecant, secant, and cotangent. Let θ?R define an angle in standard position. Choose any point Px,y on the terminal side of θ.We define as follows:189865635Six Trig Functionssinθ=yr cosθ=xr tanθ=yx, x≠0cscθ=ry, y≠0 secθ=rx, x≠0 cotθ=xy, y≠0Where r=x2+y2SOH-CAH-TOA0Six Trig Functionssinθ=yr cosθ=xr tanθ=yx, x≠0cscθ=ry, y≠0 secθ=rx, x≠0 cotθ=xy, y≠0Where r=x2+y2SOH-CAH-TOAEx 1The terminal side of an angle θ in standard position passes through the point 8, -6. Find the values of the six trig functions of angle θ.Ex 2Find the six trig function values of the angle θ in standard position, if the terminal side of θ is defined by 3x-2y=0, x≤0.Ex 3Graph y=23x. Use the graph to determine what tangent really means.Quadrantal AnglesFormed when terminal side of an angle in standard position lies along one of the axes. Occurs whenθ∈0+2kπ,π2+2kπ, π+2kπ,3π2+2kπ.2350770882654330890161163Trig Function Values at Quadrantal Anglesθ in degreesθ in radianssinθcosθtanθcotθsecθcscθEx 4 If is in quadrant III, what are the signs of the following?(a) (b) (a) Ex 5 Find each of the following:(a) (b) (c) Ex 6 Evaluate. (a) (b) cos2(-180°)+sin2(-180°)(c) 3sin2-90°-5tanπ+6sec23πNote: (Pyth Id)TRY Evaluate. (a) (b) TRYDecide whether each expressions is equal to 0, 1, or -1 or is undefined.(#94)(#96)(#100) (#102) sinn?π tan2n+1?π2 cosn?2π cscn?π Sec 1.4: Using the Definitions of the Trig FunctionsNote that cscθ=ry=1ry=1sinθcenter142875Reciprocal Identitiessinθ=1cscθ cosθ=1secθ tanθ=1cotθcscθ=1sinθ secθ=1cosθ cotθ=1tanθReciprocal Identitiessinθ=1cscθ cosθ=1secθ tanθ=1cotθcscθ=1sinθ secθ=1cosθ cotθ=1tanθEx 1Find each function value.a)b)c) (#10)tanθ, given that cotθ=4 secθ, given thatcosθ=-220 sinθ, given thatcscθ=243Signs of Function Values-16637010350500ALL STUDENTS TAKE CALCULUSEx 2Determine the signs of the trig functions of an angle in standard position with the given measure.a)b)c)d)84° -15° 6π7 8π7Ex 3Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions.a)b)c) (#36)tanθ>0,cscθ<0 sinθ>0,cscθ>0 cosθ<0,sinθ<0Find the Ranges of Trig FunctionsRanges of Trig Functionscenter7048500Ex 4Decide whether each statement is possible.a)b)c)d)cotθ=-0.999 cosθ=-1.7 cscθ=0 sinθ=-π8Ex 5Suppose that θ is in QIII and tanθ=85. Find the values of the other 5 trig functions.-90805154305Pythagorean IdentitiesFor all angles θ for which the function values are defined, the following identities hold.sin2θ+cos2θ=1 tan2θ+1=sec2θ 1+cot2θ=csc2θPythagorean IdentitiesFor all angles θ for which the function values are defined, the following identities hold.sin2θ+cos2θ=1 tan2θ+1=sec2θ 1+cot2θ=csc2θDerive Pythagorean Identitiescenter158750Quotient IdentitiesFor all angles θ for which the denominators are not zero, the following identities hold. sinθcosθ=tanθ cosθsinθ=cotθQuotient IdentitiesFor all angles θ for which the denominators are not zero, the following identities hold. sinθcosθ=tanθ cosθsinθ=cotθDerive Quotient IdentitiesEx 6Find cosθ and tanθ, given that sinθ=-23 and cosθ>0. Check your answer. Use identity.Ex 7Find sinθ and cosθ, given that cotθ=-724 and θ is in QII.Ex 8 (#74)Find the remaining 5 trig function values if cscθ=2 and θ is in QII.Sec 2.1: Trigonometric Functions of Acute AnglesAnother way to define trigonometric functions based on the ratios of the lengths of the sides of a right triangle.center31014Right-Triangle-Based Definitions of Trig FunctionssinA=yr=side opposite of Ahypotenuse cscA=ry=hypotenuseside opposite of AcosA=yx=side adjacent to Ahypotenuse secA=rx=hypotenuseside adjacent to AtanA=yx=side opposite of Aside adjacent to A cotA=xy=side adjacent to Aside opposite of ASOH-CAH-TOA0Right-Triangle-Based Definitions of Trig FunctionssinA=yr=side opposite of Ahypotenuse cscA=ry=hypotenuseside opposite of AcosA=yx=side adjacent to Ahypotenuse secA=rx=hypotenuseside adjacent to AtanA=yx=side opposite of Aside adjacent to A cotA=xy=side adjacent to Aside opposite of ASOH-CAH-TOA4394835139700ryxA0ryxA4593590889007736ABC857736ABC85Ex 1Find the sine, cosine, and tangent values for angles A and B in the figure. Note that sinA=cosB and cosA=sinB. Always true for acute angles in a right triangle.Since angles A and B are complementary and sinA=cosB, sine and cosine are cofunctions. -12382519050Cofunction IdentitiesFor any acute angle A, cofunction values of complementary angles are equal.sinA=cos90°-A secA=csc90°-A tanA=cot90°-AcosA=sin90°-A cscA=sec90°-A cotA=tan90°-A0Cofunction IdentitiesFor any acute angle A, cofunction values of complementary angles are equal.sinA=cos90°-A secA=csc90°-A tanA=cot90°-AcosA=sin90°-A cscA=sec90°-A cotA=tan90°-AEx 2Write each function in terms of its cofunction.a)sin9°b) cot76°c)csc45°Ex 3Find one solution for each equation. Assume all angles involved are acute angles. a)cotθ-8°=tan(4θ+13°) b)sec5θ+14°=csc2θ-8°Ex 4Determine whether each statement is true or false.a)tan25°<tan23°b) csc44°<csc40°Special Angles10160013462000STUDY TRIG VALUES WELL! CH 6 (INVERSE STUFF) WILL BE IMPOSSIBLE TO DO WITHOUT BEING PROFICIENT WITH IT. NEEDS TO BE NEARLY SECOND NATURE!!255968544450DegreesRadianssinθcosθtanθcscθsecθcotθ0°030°π645°π460°π390°π2180°π270°3π200DegreesRadianssinθcosθtanθcscθsecθcotθ0°030°π645°π460°π390°π2180°π270°3π2Function Values of Special Angles-138989124841Ex 5 Find the six trig function values for a 30° angle.Ex 6Evaluate. (Note: These problems are in degrees but almost everything in calculus is done in radians.)a)b)c)d)e)sin145°cos-60°cos150°tan420°sin(-150°) f)g)h)i)j)tan-45°csc-60°sec120°cot-30°csc45°Sec 2.2: Trigonometric Functions of Non-Acute Angles-10223525400DefnA reference angle for an angle θ, written θ', is the positive, acute angle made by the terminal side of angle θ and the x-axis. DefnA reference angle for an angle θ, written θ', is the positive, acute angle made by the terminal side of angle θ and the x-axis. 487426013271531649671339851607160133985-102870135255Ex 1Find the reference angle for each angle.a) 190°b) 883°c) 11π/6Ex 2Find the exact values of the 6 trig functions for 135°.Sec. 3.1 (part II) Ex 3Evaluate.a) sin-5π6b) cos5π6c) cot13π3Ex 4Evaluate sin245°+3cos2135°-2tan225°.Ex 5Evaluate each function by first expressing the function in terms of an angle between 0° and 360°.a) sin585°b) cot-930°Ex 6Find all values of θ, if θ is in the interval 0°, 360° and sinθ=-32. TRYFind the exact values of the 6 trig functions for each angle. Always start with a sketch!(a) (a) Sec 3.2: Applications of Radian MeasureFrom geometry, we know the arc lengths are proportional to the measure of their central angles.420560584455-3619567310Arc LengthThe length s of the arc intercepted on a circle of radius r by a central angle of measure θ radians is given by the product of the radius and the radian measure of the angle, or s=rθ, θ in radiansArc LengthThe length s of the arc intercepted on a circle of radius r by a central angle of measure θ radians is given by the product of the radius and the radian measure of the angle, or s=rθ, θ in radiansNOTE:In degrees, the arc length formula is s=θ3602πr=θπr180.Ex 1A circle has radius 25.60 cm. Find the length of the arc intercepted by a central angle having each of the following measures.a)b) 7π8 radians 54°Ex 2Eerie, Pennsylvania is approximately due north of Columbia, South Carolina. The latitude of Eerie is 42°N, while that of Columbia is 34°N. Find the north-south distance between the two cities. (The radius of Earth is 6400 km.)*Lattitude gives the measure of a central angle with vertex at Earth’s center, whose initial side goes through the equator and whose terminal side goes through the given location.Ex3A rope is being wound around a drum with radius 0.327 m. How much rope will be wound around the drum if the drum is rotated through an angle of 132.6°?Ex 4Two gears are adjusted so that the smaller gear drives the larger one, as shown in the figure. If the smaller gear rotates through angle of150°, through how many degrees will the larger gear rotate?-444501123953.6 in5.4 in3.6 in5.4 in29210103505Area of a SectorThe area A of a sector of a circle of radius r and central angle θ is given by the following formula.A=12r2θ, θ is in radians0Area of a SectorThe area A of a sector of a circle of radius r and central angle θ is given by the following formula.A=12r2θ, θ is in radiansProof of Area of a sector:Ex 5Find the area of a sector of a circle having radius 15.20 ft and central angle 108.0°.Sec 3.3: The Unit Circle and Circular FunctionsIn section 1.3, we defined the 6 trig functions where the domain was a set of angles in standard position. In advanced math courses, it is necessary that the domain consist of real numbers.-12382520320Circular FunctionsFor any real number s represented by a directed arc on the unit circle, sins=y coss=x tans=yx, x≠0cscs=1y, y≠0 secs=1x, x≠0 cots=xy, y≠0Circular FunctionsFor any real number s represented by a directed arc on the unit circle, sins=y coss=x tans=yx, x≠0cscs=1y, y≠0 secs=1x, x≠0 cots=xy, y≠0The unit circle is symmetric with respect to the x-axis, y-axis, and origin. We can use symmetry and trig function values in the first quadrant to evaluate trig functions in the other 3 quadrants.-32237673025Domain of Circular FunctionsSine and Cosine: -∞,∞Tangent and Secant:ss≠2n+1π2, n∈ZCotangent and Cosecant:ss≠nπ, n∈Z0Domain of Circular FunctionsSine and Cosine: -∞,∞Tangent and Secant:ss≠2n+1π2, n∈ZCotangent and Cosecant:ss≠nπ, n∈Z3277210-279Ex 1Evaluate.a)b)c)d) sin4π3 cos4π3 tan-9π4 sin11π6e)f)g) h)sec23π6 csc13π3 cot-5π6 secπ2Note:For calculator problems, be sure the setting is in radian mode.Ex 2a)Find the approximate value of s∈0,π2 if sins=0.3210.b)Find the exact value of s∈3π2,2π if tans=-33.Sec 3.4: Linear and Angular SpeedSuppose a point is moving along in a circular path. It has both linear speed and angular speed.206961712758487782128041rate=distancetime-16825069291DefnsLet P be a point on a circle of radius r moving at a constant speed. The measure of how fast P is changing is the linear speed, v.v=st, where s is the length of the arc traced by P at time tLet θ be the angle formed by an angle in standard position whose terminal side contains P. The measure of how fast θ changes is its angular speed. Angular speed, denoted ω, is given asω=θt, where θ is in radiansNote: v=st=rθt=rω0DefnsLet P be a point on a circle of radius r moving at a constant speed. The measure of how fast P is changing is the linear speed, v.v=st, where s is the length of the arc traced by P at time tLet θ be the angle formed by an angle in standard position whose terminal side contains P. The measure of how fast θ changes is its angular speed. Angular speed, denoted ω, is given asω=θt, where θ is in radiansNote: v=st=rθt=rωEx 1Suppose that P is on a circle with radius 15 in. and ray OP is rotating with angular speed π12 radians per second.a)Find the angle generated by P in 10 sec.b)Find the distance traveled by P along the circle in 10 sec.c)Find the linear speed of P in inches per second.Ex 2A belt runs a pulley of radius 5 in. at 120 revolutions per minute. a)Find the angular speed of the pulley in radians per second.b)Find the linear speed of the belt in inches per second.Ex 3A satellite traveling in a circular orbit approximately 1800 km above the surface of Earth takes 2.5 hrs to make an orbit. (The Earth’s radius is 6400 km.)a)Approximate the linear speed of the satellite in kilometers per hour. A sketch will help!b)Approximate the distance the satellite travels in 3.5 hrs.Sec 2.4: Solving Right TrianglesMost values obtained for trig applications are not exact. We will round our final answers using significant digits. That is, we round the final answer to the same number of significant digits as the number with the least number of significant digits.In the following numbers, the sig figs are identified in color/bold.408 21.5 17.00 6.700 0.0025 0.09840 74004468611283241c35.9 km31°40'bACBc35.9 km31°40'bACBEx 1 (#12)Solve the right triangle. That is, find the measures of all angles and sides of the triangle. Provide angles answers in degrees and minutes. (Round using sig figs.)Angles of Elevation or DepressionIn applications of right triangles, the angle of elevation from point X to Y (above X) is the acute angle formed by XY and a horizontal ray with endpoint X. The angle of depression from point X to Y (below X) is the acute angle formed by XY and a horizontal ray with endpoint X.Both angles are measured between the line of sight and a horizontal line (‘x-axis’).2611582117648Angle of DepressionHorizontalXYAngle of DepressionHorizontalXY-22167313739Angle of ElevationHorizontalXYAngle of ElevationHorizontalXYEx 2 (#56)The length of the shadow of a flagpole 55.20 ft tall is 27.65 ft. Find the angle of elevation of the sun to the nearest hundredth of a degree.Section 2.4Name: _________________________________Date :_____________________Practice Word Problems on Right TrianglesSolve the following problems. You need to attach a sketch for each problem and label clearly.1) A guy wire is attached to a 100-ft tower that is perpendicular to the ground. The wire makes an angle of 55 with the ground. What is the length of the wire?A fire is sighted from a fire tower in Wayne National Forest. The ranger found that the angle of depression to the fire is 22. If the tower is 75 meters tall, how far is the fire from the base of the tower?A surveyor is 100 meters from a building. He finds that the angle of elevation to the top of the building is 23. If the surveyor’s eye level is 1.55 meters above the ground, find the height of the building.From the top of the lighthouse, the angle of depression to a buoy is 25. If the top of the lighthouse is 150 feet above sea level, find the distance from the buoy to the foot of the lighthouse.5) A ship travels 50 km on a bearing of 27, the travels on a bearing of 117 for 140 km. Find the distance traveled from the starting point to the ending point.Sec 2.5: Further Applications of Right TrianglesFor full credit on quizzes/exams an accurate, correctly labeled sketch must be drawn. Use your sketch to check reasonability of your answer.Bearing is an important concept used in navigation and is measured in a clockwise direction from due north.Sample (true) bearings:33°, 164°, 229°, 304°We may also express bearings by using a north-south line and using an acute angle to show the direction, either east or west, from the line.Sample (conventional) bearings:N42°E, S31°E, S40°W, N52°WEx 1 (#12)An observer of a radar station is located at the origin. Find the bearing of an airplane located at 2, 2. Express the bearing using both methods.Ex 2 (#18)Two ships leave a port at the same time. The first ship sails on a bearing of 52° at 17 knots and the second on a bearing of 322° at 22 knots. How far apart are they after 2.5 hr?Ex 3 (#34)Debbie Glockner-Ferrari, a whale researcher, is watching a whale approach directly toward a lighthouse. When she first begins watching the whale, the angle of depression to the whale is 15°50'. Just as the whale turns away from the lighthouse, the angle of depression is 35°40'. If the height of the lighthouse is 68.7 m, find the distance traveled by the whale as it approached the lighthouse.Sec 7.1: Oblique Triangles and the Law of SinesIn section 2.4, we solved right triangles. We now extend the concept to all triangles.Congruence Axiomscenter76352Side-Angle-SideSASIf two sides and the included angle of one triangle are equal, respectively, to two sides and the included angle of a second triangle, then the triangles are congruent.Angle-Side-AngleASAIf two angles and the included side of one triangle are equal, respectively, to two angles and the included side of a second triangle, then the triangles are congruent.Side-Side-SideSSSIf three sides of one triangle are equal, respectively, to three sides of a second triangle, then the triangles are congruent.Side-Angle-SideSASIf two sides and the included angle of one triangle are equal, respectively, to two sides and the included angle of a second triangle, then the triangles are congruent.Angle-Side-AngleASAIf two angles and the included side of one triangle are equal, respectively, to two angles and the included side of a second triangle, then the triangles are congruent.Side-Side-SideSSSIf three sides of one triangle are equal, respectively, to three sides of a second triangle, then the triangles are congruent.Whenever SAS, ASA, or SSS is given, the triangle is unique.An oblique triangle is a triangle that is not a right triangle. Information sufficient to solve an oblique triangle:(1) one side(2) any other two measuresData Required for Solving Oblique Triangles29972074803One SideTwo AnglesSAA or ASACase 1Two SidesOne Angle not included between the two sidesThis may lead to more than one triangleSSACase 2Two SidesOne Angle included between the two sidesSASCase 3Three SidesSSSCase 4Use Law of SinesSec 7.1 & 7.2Use Law CosinesSec 7.3One SideTwo AnglesSAA or ASACase 1Two SidesOne Angle not included between the two sidesThis may lead to more than one triangleSSACase 2Two SidesOne Angle included between the two sidesSASCase 3Three SidesSSSCase 4Use Law of SinesSec 7.1 & 7.2Use Law CosinesSec 7.3Derive the Law of Sines106172-189992Law of SinesIn any triangle ABC, with sides a, b, and c, asinA=bsinB=csinCNote:The ratio above is the diameter of the circumscribed circle of the triangle. See Exercise 53.Alternative form:sinAa=sinBb=sinCc0Law of SinesIn any triangle ABC, with sides a, b, and c, asinA=bsinB=csinCNote:The ratio above is the diameter of the circumscribed circle of the triangle. See Exercise 53.Alternative form:sinAa=sinBb=sinCcWe can use the same method used to derive the Law of Sines to derive formulas for the area of a rectangle (with an unknown height).-10922029845Area of a Triangle (SAS)In any triangle ABC, the area A is given by the following formulas.A=12bcsinA A=12absinC A=12acsinBArea of a Triangle (SAS)In any triangle ABC, the area A is given by the following formulas.A=12bcsinA A=12absinC A=12acsinB43815009271000Ex 1 (#4)Find the length of side a.Ex 2 (#12)Solve the triangle.B=38°40', a=19.7 cm, C=91°40'462280022796500Ex 3 (#26)To determine the distance RS across a deep canyon, Rhonda lays off a distance TR=582 yd. She then finds that T=32°50' and R=102°20'. Find RS. Ex 4 (#30)Standing on one bank of a river flowing north, Mark notices a tree on the opposite bank at a bearing of 115.45°. Lisa is on the same bank as Mark, but 428.3 m away. She notices that the bearing of the tree is 45.47°. The two banks are parallel. What is the distance across the river?Ex 5. The angle of depression from the top of a building to a point on the ground is . How far is the point on the ground from the top of the building if the building is 252 m high?Ex 6 Two docks are located on an east-west ling 2,587 feet apart. From dock A, the bearing of a coral reef is . From dock B, the bearing of the coral reef is . Find the distance from dock A to the coral reef.526669020256500Ex 7 (#34)Three atoms with atomic radii of 2.0, 3.0, and 4.5 are arranged as in the figure. Find the distance between the centers of atoms A and C.495935030226000Ex 8 (#40)Find the area of the triangle using the formula A=12bh, and then verify that the formula A=12absinC gives the same result.478409023368000Ex 9 (#55)Several of the exercises on right triangle applications involve a figure similar to the one shown Use the law of sines to obtain x in terms of α, β, and d.Sec 7.3: The Law of Cosines-11704379070Triangle Side Length RestrictionIn any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. (Sum of two shortest sides must be greater than the longest side.)0Triangle Side Length RestrictionIn any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. (Sum of two shortest sides must be greater than the longest side.)Ex 1A triangle has base of length 13 and the other two sides are equal in length. If the lengths are integers, what is the shortest possible length of the unknown side? Derive the Law of Cosines(using the distance formula)431165-285750Law of CosinesIn any triangle ABC with sides a, b, and c, the following hold.a2=b2+c2-2bccosAb2=a2+c2-2accosBc2=a2+b2-2abcosCLaw of CosinesIn any triangle ABC with sides a, b, and c, the following hold.a2=b2+c2-2bccosAb2=a2+c2-2accosBc2=a2+b2-2abcosCSee “Suggested Procedure for Solving Oblique Triangles” on page 310.Ex 2Determine whether SAA, ASA, SSA, SAS, or SSS is given. What law would you use?#1#2#3#4a, b, and C A, C, and c a, b, and A a, B, and C#5#6#7#8A, B, and c a, c, and A a, b, and c b, c, and A423550114376θ113θ113Ex 3 (#12)Find the measure of θ. (No calculator.)-300355153670A4108BCA4108BCEx 4Solve each triangle.(#16)(#34)B=168.2°a=15.1 cmc=19.2 cmEx 5 (#40)Points X and Y are on opposite sides of a ravine. From a point third point Z, the angle between the lines of sight to X and Y is 37.7°. If XZ is 153 m long and YZ is 103 m long, find XY.Ex 6 (#44)An airplane flies 180 mi from point X at a bearing of 125°, and then turns and flies at a bearing of 230° for 100 mi. How far is the plane from point X.441833022923500Ex 7 (#51)A weight is supported by cables attached to both ends of a balance beam, as shown in the figure. What angles are formed between the beams and cables?409638575565000Ex 8 (#57)Surveyors are often confronted with obstacles, such as trees, when measuring the boundary of a lot. One technique used to obtain an accurate measurement is the so-called triangulation method. In this technique, a triangle is constructed around the obstacle and one angle and two sides of the triangle are measured. Use this technique to find the length of the property line (the straight line between the two markers) in the figure.-1524031572Heron’s Formula (SSS)If a triangle has side lengths a, b, and c with semiperimeter s=12a+b+c,then the area A of the triangle is given by the formula:A=ss-as-bs-c00Heron’s Formula (SSS)If a triangle has side lengths a, b, and c with semiperimeter s=12a+b+c,then the area A of the triangle is given by the formula:A=ss-as-bs-cEx 9 (#66)Find the area of the triangle ABC. a=22 in., b=45 in., c=31 in.Sec 7.2: The Ambiguous Case of the Law of SinesRecap: To solve a triangle, we need to to find ____________________________________________.To solve a triangle, we need to be given AT LEAST _________________________________________.Six possible cases of 3 given measurements:Angle-Angle-AngleOne side and any two anglesTwo sides and an included angleSide-Side-SideTwo sides and a non-included angleWhy is SSA ambiguous?Let θ and a be fixed and assume θ is acute. -635137160θaθaEx 1 (#8)Determine the number of triangles possible with the given parts.B=54°, c=28, b=23Ex 2Find the unknown angles in triangle ABC for each triangle that exists.(#18) C=82.2°, a=10.9 km, c=7.62 kmEx 3Solve the triangle that exists.(#22) C=52.3°, a=32.5 yd, c=59.8 ydEx 4(#30) A=51.20°, c=7986 cm, a=7208 cmEx 5 (#36)440372537401500The angle of elevation from the top of a building 45.0 ft high to the top of a nearby antenna tower is 15°20'. From the base of the building, the angle of elevation of the tower is 29°30'. Find the height of the tower.Ex 6 (#38) A pilot flies her plane on a heading of 35°00' from point X to point Y which is 400 mi from X. Then she turns and flies on a heading of 145°00' to point Z, which is 400 mi from her starting point X. What is the heading of Z from X, and what is the distance YZ?Ex 7 (#40) Use the law of sines to prove the statement is true for any triangle ABC, with corresponding sides a, b, and c.a-ba+b=sinA-sinBsinA+sinBSummary of Solving a TriangleOblique TriangleWhat do I do?Sec 4.1: Graphs of the Sine and Cosine FunctionsGraph of y=sinx2070100476250032835854699000xy=sinxx,y000, 0π612π6,12π21π2, 15π6125π6,12π0π, 07π6-127π6,-123π2-13π2,-111π6-1211π6,-122π02π, 033864557302500Ex 1Graph each function.y=3sinxy=3sinx+1214312517780000y=sin2x-89535-170815DefnA periodic function is a function f such that fx=fx+np for every real number x in the domain of f, every integer n, and some positive real number p. The least positive value of p is the period of the function.Periodic functions “repeat” with a regular pattern. DefnA periodic function is a function f such that fx=fx+np for every real number x in the domain of f, every integer n, and some positive real number p. The least positive value of p is the period of the function.Periodic functions “repeat” with a regular pattern. -8953531808Properties of the Sine Function1) domsinx=2) rngsinx=3) The sine function is an ______ function, as the symmetry of the graph with respect to the origin indicates. Moreover, sin-x=________ ?x∈domsinx.4) The sine function is periodic, with period ______.5) The x-intercepts occur at x=0, ±π, ±2π, ±3π,…6) The y-intercept is 0,0.7) The maximum value is 1 and occurs at x=…-3π2,π2,5π2,9π2,…8) The minimum value is -1 and occurs at x=…-π2,3π2,7π2,11π2,…0Properties of the Sine Function1) domsinx=2) rngsinx=3) The sine function is an ______ function, as the symmetry of the graph with respect to the origin indicates. Moreover, sin-x=________ ?x∈domsinx.4) The sine function is periodic, with period ______.5) The x-intercepts occur at x=0, ±π, ±2π, ±3π,…6) The y-intercept is 0,0.7) The maximum value is 1 and occurs at x=…-3π2,π2,5π2,9π2,…8) The minimum value is -1 and occurs at x=…-π2,3π2,7π2,11π2,…32950153429000Ex 2Graph each function.y=cosxy=-cosx+1235204012636500y=2cos3x-125095-128905Properties of the Cosine Function1) domcosx=2) rngcosx=3) The cosine function is an ______ function, as the symmetry of the graph with respect to the y-axis indicates. Moreover, cos-x=________ ?x∈domcosx.4) The cosine function is periodic, with period ______.5) The x-intercepts occur at x=±π2,±3π2, ±5π2,…6) The y-intercept is 0,1.7) The maximum value is 1 and occurs at x=0,±2π., ±4π, ±6π,…8) The minimum value is -1 and occurs at x=±π, ±3π, ±5π, …0Properties of the Cosine Function1) domcosx=2) rngcosx=3) The cosine function is an ______ function, as the symmetry of the graph with respect to the y-axis indicates. Moreover, cos-x=________ ?x∈domcosx.4) The cosine function is periodic, with period ______.5) The x-intercepts occur at x=±π2,±3π2, ±5π2,…6) The y-intercept is 0,1.7) The maximum value is 1 and occurs at x=0,±2π., ±4π, ±6π,…8) The minimum value is -1 and occurs at x=±π, ±3π, ±5π, …-12446090805DefnThe amplitude of a periodic function is half the difference between the maximum and minimum values. The graph of y=asinx or y=acosx, with a≠0, will have the same shape as the graph of y=sinx or y=cosx, respectively, except with range -a, a. The amplitude is ______.00DefnThe amplitude of a periodic function is half the difference between the maximum and minimum values. The graph of y=asinx or y=acosx, with a≠0, will have the same shape as the graph of y=sinx or y=cosx, respectively, except with range -a, a. The amplitude is ______.Calculate the amplitude of y=3sinx and y=2cosx. Provide a sketch of each graph.238252017208500Ex 3Make a conjecture about the graphs of y=sinx and y=cosx.Conjecture:sinx=Note:Because of this relationship, the graphs of y=Asinωx or y=Acosωx are referred to as sinusoidal graphs.Ex 4Find the period of y=sinx, y=cosx, y=-sin2x, and y=2cos3x.27940149225PeriodFor b>0, the graph of y=sinbx will resemble that of y=sinx, but with period 2πb. Similarly for y=cosbx and y=cosx.PeriodFor b>0, the graph of y=sinbx will resemble that of y=sinx, but with period 2πb. Similarly for y=cosbx and y=cosx.Ex 5Graph each function over a two-period interval. Give the period and amplitude.(#38’) y=23sinπ4x fx=-2cos13x-3937016129000334073516129000Ex 6Write the function that describes each graph. Assume no phase (horizontal) shift.372554511239500356870762000(a) (b)68580014160541433759525 4305306540500388810511430000(c)(d)904875889042291001320805245105588000(e)38290501270000(f)9239251225554314825825552641512065008858251361440392557010604500(g)39376357620Sec 4.2: Translations of the Graphs of the Sine and Cosine FunctionsRecall From Algebra:-4826066675DefnWith circular functions, a horizontal translation is called a phase shift. In the function fx, the input value is called the argument.0DefnWith circular functions, a horizontal translation is called a phase shift. In the function fx, the input value is called the binations of Translations y=asinbx-d+c or y=acosbx-d+c, where b>0Ex 1 Find the amplitude, the period, any vertical translation, and any phase shift of the graph of each function. (b) 335724511430001295401143000(#26)(#30)c y=3sinx+π2 (d) fx=-cosπx-1360325174625006978655334000-4635567945000333375067945000(#48)(#52)e fx=1-23sin34x (f) y=1+23cos12xEx 2Match the function with its graph.A) y=sinx-π2 B) y=sinx+π2 C) y=cosx-π2 D) y=cosx+π2E) y=sinx+2 F) y=sinx-2 G) y=cosx+2 H) y=cosx-239052503175003600454064000(a)(b)_____________________________________________39052503810000431800698500(c)(d)_____________________________________________399478512065004552951143000(e)(f)_____________________________________________393509536195004502157683500(g)(h)_____________________________________________Sec 4.3: Graphs of the Tangent and Cotangent FunctionsBecause the tangent function has period π, we only need to determine the graph over some interval of length π. (The rest of the graph will consist of repetitions over some interval of length π.) Because the tangent function is not defined at …,-3π2, -π2,π2,3π2,…, we will concentrate on the interval -π2,π2.27228801714500xy=tanxx,y-π3-3≈-1.73-π3, -3-π4-1-π4,-1-π6-33≈-0.58-π6, -33000, 0 π633≈0.58π6, 33π41π4,1π33≈1.73π3, 3What happens when x-values approach π/2?That is, what is limx→π2-tanx? See tablexsinxcosxy=tanxπ3≈1.0532123≈1.731.50.99750.070714.11.570.99990.0007961255.81.57070.99990.00009610,381π2≈1.570810UndefinedThe tangent function has a vertical asymptote everywhere it’s undefined, that is, when cosx=0. On -π2,π2, it has a vertical asymptote at _____________________________.-228600161405Properties of the Tangent Function1) domtanx=2) rngtanx=3) The tangent function is an ______ function, as the symmetry of the graph with respect to the origin indicates. Moreover, tan-x=________ ?x∈domtanx.4) The tangent function is periodic, with period ______.5) The x-intercepts occur at x=0, ±π, ±2π, ±3π,…6) The y-intercept is 0,0.7) Vertical asymptotes occur at x=…±π2,±3π2,…00Properties of the Tangent Function1) domtanx=2) rngtanx=3) The tangent function is an ______ function, as the symmetry of the graph with respect to the origin indicates. Moreover, tan-x=________ ?x∈domtanx.4) The tangent function is periodic, with period ______.5) The x-intercepts occur at x=0, ±π, ±2π, ±3π,…6) The y-intercept is 0,0.7) Vertical asymptotes occur at x=…±π2,±3π2,…Ex 1Graph each. fx=2tanx y=-3tan2x-157480876300032226258763000We define the cotangent function as we did the tangent function. The period of y=cotx is also π and because cotangent is not defined for integer multiples of π, we will concentrate on the interval 0,π. 325310514478000xy=cotxπ63≈1.73π41π333≈0.58π202π3-33≈-0.583π4-15π6-3≈-1.73limx→0+cotx limx→π-cotxThe cotangent function has a vertical asymptote everywhere it’s undefined, that is, when sinx=0. On 0,π, it has a vertical asymptote at _____________________________.Ex 2Graph each.192087531242000429133031432500fx=1-12tan2x fx=2cotx-1 f(x)=1-cot12x-4533906350000-76200-146685Properties of the Cotangent Function1) domcotx=2) rngcotx=3) The cotangent function is an ______ function, as the symmetry of the graph with respect to the origin indicates. Moreover, cot-x=________ ?x∈dom f.4) The cotangent function is periodic, with period ______.5) The x-intercepts occur at x=±π2, ±3π2,±5π2,…6) The y-intercept is ________.7) Vertical asymptotes occur at x=0, ±π, ±2π, ±3π,…00Properties of the Cotangent Function1) domcotx=2) rngcotx=3) The cotangent function is an ______ function, as the symmetry of the graph with respect to the origin indicates. Moreover, cot-x=________ ?x∈dom f.4) The cotangent function is periodic, with period ______.5) The x-intercepts occur at x=±π2, ±3π2,±5π2,…6) The y-intercept is ________.7) Vertical asymptotes occur at x=0, ±π, ±2π, ±3π,…Ex 3(#43’)If c is any number, how many solutions does the equaton c=tanx have in the interval -2π,2π? -2π, 2π?Ex 4Write the function for each graph. Assume no phase shift. 446405571500(a)3879215889000 (b)________________________________________________3974465171450005657852603500(c)(d)______________________________________________________62039513906500(e)________________________________Sec 4.4: Graphs of the Secant and Cosecant FunctionsThe secant and cosecant functions, sometimes referred to as reciprocal functions, are graphed by making use of the reciprocal identities.cscx=1sinx secx=1cosxNote that secant is an even function. Thus, sec-x=secx ?x∈domsecx.31908756667500xy=secx01±π6233≈1.2±π42≈1.4±π32±2π3-2±3π4-2≈-1.4±5π6-233≈-1.2±π-131902405842000xy=cscxxy=cscxπ62-π6-2π42≈1.4-π4-2≈-1.4π3233≈1.2-π3-233≈-1.2π21-π2-12π3233≈1.2-2π3-233≈-1.23π42≈1.4-3π4-2≈-1.45π62-5π6-2Ex 1Graph each function over a one period interval. a y=-2cscx-π2 (b) y=3sec2x-π+2-28956026924000309054526924000Ex 2Write the function for each graph.-360218245630 (b)(c)Sec 4.5: Harmonic MotionSimple harmonic motion is an important concept for those studying engineering or physics.-1778014478000Idea: A weight is attached to a coiled spring. It is pulled down a distance of 20 cm from its equilibrium position and released. The time for one complete oscillation is 4 seconds.VISUAL:-69215-174567Simple Harmonic MotionThe position of a point oscillating about an equilibrium position at time t is modeled by either st=acosωt or st=asinωt, where a and ω are constants, with ω>0. The amplitude of the motion is a, the period is 2π/ω, and the frequency is ω/2π oscillations per minute. (The number of cycles per unit of time is called the frequency and is equal to the reciprocal of the period.)0Simple Harmonic MotionThe position of a point oscillating about an equilibrium position at time t is modeled by either st=acosωt or st=asinωt, where a and ω are constants, with ω>0. The amplitude of the motion is a, the period is 2π/ω, and the frequency is ω/2π oscillations per minute. (The number of cycles per unit of time is called the frequency and is equal to the reciprocal of the period.)Ex 1(#4)A weight on a spring has initial position s0=-4 and period P=1.2 sec. (a)Find a function s given by st=acosωt that models the displacement of the weight.(b)Evaluate s1. Is the weight moving upward, downward, or neither when t=1? Support your results graphically or numerically.Ex 2(#6)A note on the piano has frequency F=110. Suppose the maximum displacement at the center of the piano wire is given by s0=0.11. Find constants a and ω so that the equation st=acosωt models this displacement. Graph s in the viewing window 0, 0.05 by -0.3, 0.3.38017458890000519366540513000Ex 3(#10)An object is attached to a coiled spring as in the figure given. It is pulled down a distance of 6 units from its equilibrium position and then released. The time for one complete oscillation is 4 sec.(a)Give an equation that models the position of the object at time t.(b)Determine the position at t=1.25 sec.(c)Find the frequency.Ex 4(#14)The period in seconds of a pendulum of length L in feet is given by:P=2πL32 .How long should the pendulum be to have a period of 1 sec?Ex 5(#18)The position of a weight is attached to a spring is st=-4cos10t inches after t seconds. a)What is the maximum height that the weight rises above the equilibrium position?b)What are the frequency and period?c)When does the weight first reach its maximum height?d)Calculate and interpret s1.466.Chapter 4 Trig Graphing ReviewName :______________________________Date :___________________1. Determine the period of each function.(a) (b) 341249086995001143002984500(c) (d) 3412490914400011430034290002. Determine the amplitude, phase shift, and range for each function.(a) (b) 341249086995001143002984500(c) (d) 3412490914400011430034290003. Graph each of the following trigonometric functions. Graph at least one period. Clearly label the intervals, any asymptotes, etc. (a) (b) (c) (d) (e) (f) (g) (h) 4. Write the function that describes each graph. Assume no phase (horizontal) shift.423799016764000-3124206413500280670-190500(a) (b) (c)436181512573000218313012573000-44453873500-40747951657350020675602857500-312420374650004297045381000(d) (e) (f) 436181512573000218313012573000-44453873500440690374650032778701098550035560005537200061214062992000 (g)(h) 5. Write the function for each graph. Assume no phase shift.-45847012700000181038520637500421386024892000 (a)43256201085850063538100020916907048500(a)(b) (c) 6. Match the function with its graph.A) y=sinx-π2 B) y=sinx+π2 C) y=cosx-π2 D) y=cosx+π2E) y=sinx+2 F) y=sinx-2 G) y=cosx+2 H) y=cosx-2 46005756985000-24955515113000933451778000 -35229808191500-149415514605000 (a) (b)-189230102870001222375184150021399514986000(c)78041510858500-1473200156845003060701333500 (d) (e)24980905842000-94615990600036271206794500-396494046736000-144843546672500105664016065500Sec 5.1: Fundamental IdentitiesAn identity is an equation that is satisfied by every value in the domain of its variable. We will be verifying trig identities in section 5.2.Below are identities that must be memorized perfectly for hw, quizzes, and exams. Note that numerical values can be used to help check whether or not an identity was recalled correctly.-241300104140Fundamental IdentitiesReciprocal Identitiescotθ=1tanθ secθ=1cosθ cscθ=1sinθQuotient Identitiestanθ=sinθcosθ cotθ=cosθsinθPythagorean Identitiessin2θ+cos2θ=1 tan2θ+1=sec2θ 1+cot2θ=csc2θNegative-Angle Identitiessin-θ=-sinθ cos-θ=cosθ tan(-θ)=-tanθcsc-θ=-cscθ sec-θ=secθ cot(-θ)=-cotθ00Fundamental IdentitiesReciprocal Identitiescotθ=1tanθ secθ=1cosθ cscθ=1sinθQuotient Identitiestanθ=sinθcosθ cotθ=cosθsinθPythagorean Identitiessin2θ+cos2θ=1 tan2θ+1=sec2θ 1+cot2θ=csc2θNegative-Angle Identitiessin-θ=-sinθ cos-θ=cosθ tan(-θ)=-tanθcsc-θ=-cscθ sec-θ=secθ cot(-θ)=-cotθ502539027940Warning:tan θ=-34 does NOT imply that sin θ=-3 and cos θ=400Warning:tan θ=-34 does NOT imply that sin θ=-3 and cos θ=4Ex 1 (#16)Find sinθ if secθ=72 and tanθ<0.Ex 2 (#38)Find the remaining five trig functions if cosθ=-14 and sinθ>0.Ex 3Write each expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression and all functions are of θ only.a) (#70)1+cotθcotθb) (#76)c) (#82)sinθ-cosθcscθ+secθ -sec2-θ+sin2-θ+cos2-θEx 4 (#86)Let cscx=-3. Find all possible values of sinx+cosxsecxInclude Factoring reviewSec 5.2: Verifying Trigonometric IdentitiesOne of the skills required in more advanced mathematics, especially calculus, is the ability to use identities to write expressions in alternative forms.Warning: Techniques used to solve equations, such as adding the same term to each side, and multiplying each side by the same term. To avoid the temptation, one strategy is to work with only ONE side and rewrite it until it matches the other side. Your proofs will be graded on not only correctness but presentation as well. If both sides of an identity is equally complex, the identity can be verified by working independently on the left side and on the right side, but these steps are considered scratch work. You must rewrite the proof using the fact that each step (of one side) of your scratch work is reversible.Ex 1 (#12)Perform each indicated operation and simplify the result so that are no quotients.1sinα-1-1sinα+1Ex 2Factor each trig expression. (Simplify if possible.)a) (#16)b) (#18)tanx+cotx2-tanx-cotx2 4tan2β+tanβ-3c) (#22)d)sin3α+cos3α 3cot3x+15cot2x-12cotx-60Ex 3 Each expression simplifies to a constant, a single function, or a power of a function. Use the fundamental identities to simplify each expression.a) (#32)b) (#34)1tan2α+cotαtanα 1-1sec2xEx 4Verify that each trig equation is an identity.a) (#42)sin2α+tan2α+cos2α=sec2αb) (#48)c) (#66)secα-tanα2=1-sinα1+sinα 1+sinθ1-sinθ-1-sinθ1+sinθ=4tanθsecθd) (#70) 1+sinx+cosx2=21+sinx1+cosxe) (#76)sin2x1+cotx+cos2x1-tanx+cot2x=csc2xf) (#78)g)sin3θ+cos3θ=cosθ+sinθ1-cosθsinθ 1-sinθ1+sinθ=secθ-tanθ2Sec 5.3: Sum and Difference Identities for CosineDoes cosA-B=cosA-cosB? Why or why not?Derive the formula for cosA-B. (Sum Identity for Cosine) Use the result to find the Difference Identity for Cosine.293370022860Cofunction Identitiescosπ2-θ=sinθ cotπ2-θ=tanθsinπ2-θ=cosθ secπ2-θ=cscθtanπ2-θ=cotθ cscπ2-θ=secθ00Cofunction Identitiescosπ2-θ=sinθ cotπ2-θ=tanθsinπ2-θ=cosθ secπ2-θ=cscθtanπ2-θ=cotθ cscπ2-θ=secθ-1460533655Cosine of a Sum or DifferencecosA+B=cosAcosB-sinAsinBcosA-B=cosAcosB+sinAsinB00Cosine of a Sum or DifferencecosA+B=cosAcosB-sinAsinBcosA-B=cosAcosB+sinAsinBUse a cosine identity to derive a cofunction identity below.Ex 1Find each exact value.a) (#12)b) (#16)cosπ12 cos7π9cos2π9-sin7π9sin2π9Ex 2 (#40)Find one angle that satisfies the equation.cosθ=sinθ4+3°Ex 3(#48)Write the expression as a function of θ. cos90°+θEx 4 (#56)Find coss+t and coss-t.coss=24 andsint=-56, s and t in quadrant IVEx 5 (#58)Determine if the statement is true or false.cos-24°=cos16°-cos40°Ex 6Verify that each equation is an identity.(#70)1+cos2x-cos2x=cos2x Sec 5.4: Sum and Difference for Sine and TangentDerive the Sum Identity for Sine. Next, use the result to find the Difference Identity for Sine.4109085125730Sine of a Sum or DifferencesinA+B=sinAcosB+cosAsinBsinA-B=sinAcosB-cosAsinBTangent of a Sum or DifferencetanA+B=tanA+tanB1-tanAtanBtanA-B=tanA-tanB1+tanAtanB00Sine of a Sum or DifferencesinA+B=sinAcosB+cosAsinBsinA-B=sinAcosB-cosAsinBTangent of a Sum or DifferencetanA+B=tanA+tanB1-tanAtanBtanA-B=tanA-tanB1+tanAtanBDerive the Sum Identity for Tangent. Ex 1 Find each exact value.(#16) (#18)sin-5π12 tan-7π12Ex 2 Write each function as a single function of x or θ.(#28)(#36)cosθ-30° sin3π4-xEx 3 (#48) Given:coss=-1517 and sint=45, s in QII and t in QIFind (a) sins+t(b) tans+t(c) the quadrant of s+tEx 4Find each exact value.(#56)tan165° sin-13π12Ex 5Verify each identity.(#62) PPsinx+y+sinx-y=2sinxcosy(#66)sins+t cosscost=tans+tantSec 5.5: Double-Angle IdentitiesDerive the double-angle identity for cosine.cos2A=2222578105Double-Angle Identitiescos2A=cos2A-sin2A cos2A=1-2sin2Acos2A=2cos2A-1 sin2A=2sinAcosA tan2A=2tanA1-tan2AProduct-to-Sum IdentitiescosAcosB=12cosA+B+cosA-BsinAsinB=12cosA-B-cosA+B sinAcosB=12sinA+B+sinA-B cosAsinB=12sinA+B-sinA-B Sum-to-Product IdentitiessinA+sinB=2sinA+B2cosA-B2sinA-sinB=2cosA+B2sinA-B2cosA+cosB=2cosA+B2cosA-B2cosA-cosB=-2sinA+B2sinA-B200Double-Angle Identitiescos2A=cos2A-sin2A cos2A=1-2sin2Acos2A=2cos2A-1 sin2A=2sinAcosA tan2A=2tanA1-tan2AProduct-to-Sum IdentitiescosAcosB=12cosA+B+cosA-BsinAsinB=12cosA-B-cosA+B sinAcosB=12sinA+B+sinA-B cosAsinB=12sinA+B-sinA-B Sum-to-Product IdentitiessinA+sinB=2sinA+B2cosA-B2sinA-sinB=2cosA+B2sinA-B2cosA+cosB=2cosA+B2cosA-B2cosA-cosB=-2sinA+B2sinA-B2Ex 1Find the values of the sine and cosine functions for each angle measure.a) (#12)b) (#14)2θ, givencosθ=35 andsinθ>0 θ, givencos2θ=34 and θ terminates in QIIIEx 2Verify each identity.(#30)(#32) cos2x=1-tan2x1+tan2x cotA-tanAcotA+tanA=cos2A(#36)cotθtanθ+π-sinπ-θcosπ2-θ=cos2θEx 3(#38)(#40) 2tan15°1-tan215° 1-2sin22212°(#46)(#48)18sin29.5°cos29.5° cos22x-sin22xEx 4 (#50)Express cos3x as a trig function of x.Ex 5 (#60)Write as a sum or difference of trig functions.5cos3xcos2xEx 6 (#66)Write as a product of trig functions.sin102°-sin95°Chapter 5 Identity ReviewName:__________________________Date:_____________Verify each of the following identities:3105150363855001. 2. 310515086360002895600523240003. 4. 3352800143510005. 281940086360006. 239077586360007. 281940086360008. 28194008636000Sec 5.6: Half-Angle IdentitiesDerive the half-angle identity for sine, cosine and tangent.sinA2=6731064770Half-Angle IdentitiesChoose the correct sign based on the function under consideration and the quadrant of A/2. cosA2=±1+cosA2 sinA2=±1-cosA2 tanA2=±1-cosA1+cosA tanA2=sinA1+cosA tanA2=1-cosAsinANOTE: The last two formulas have the advantage of not requiring a sign choice.00Half-Angle IdentitiesChoose the correct sign based on the function under consideration and the quadrant of A/2. cosA2=±1+cosA2 sinA2=±1-cosA2 tanA2=±1-cosA1+cosA tanA2=sinA1+cosA tanA2=1-cosAsinANOTE: The last two formulas have the advantage of not requiring a sign choice.Ex 1 (#12)Evaluate.sin195°Ex 2Find each.a) (#24)cosx2, givencotx=-3, withπ2<x<πb) (#26)cotθ2, given thattanθ=-52, with 90°<θ<180°c) (#30)sinx, givencos2x=23, with π<x<3π2Ex 3Write as a single trig function.(#38)(#40)(#44)sin158.2°1+cos158.2° ±1+cos20α2 ±1-cos3θ52 Ex 4Verify each identity.(#48)(#52)sin2x2sinx=cos2x2-sin2x2 cosx=1-tan2x2 1+tan2x2 Ex 51528445158750tanA2=sinA1+cosA00tanA2=sinA1+cosA46589950tanA2=1-cosAsinA00tanA2=1-cosAsinA(#53) Use the half-angle identity to derive the equivalent identityby multiplying the numerator and the denominator by 1-cosA (the ‘conjugate’).Sec 6.1: Inverse Circular FunctionsTry Problem:Given . (a) Is this a function? (b) Is it a one-to-one function? (c) Find its inverse.Fundamental Topics of Trig:Graphs, identities, inverse functions and notation, solving equations. All require proficiency with trig evaluation.Review-8763037465DefnsA function is a relation (a set of ordered pairs) where each x-value corresponds to exactly ONE y-value. A one-to-one (1-1) function is a function where each y-value corresponds to exactly ONE x-value.If f is a function, the inverse of f is a relation consisting of all ordered pairs y,x where x,y∈f. If f is 1-1, then its inverse is a function. The inverse function, denoted f-1, is defined asf-1=y,xx,y∈fWarning! f-1≠1fHorizontal Line Test (HLT)A function is one-to-one if every horizontal line interests the graph of the function at most once.4 Biggies of Inverse Functions'x' and 'y' values switch placesinverses reflect in the line y=x ff-1x=x and f-1fx=x The inverse is a function iff fx is one-to-one. How to Find the Inverse of a 1-1 FunctionStep 1: Replace fx with yStep 2: x?y (interchange x and y)Step 3: Solve for y.Step 4: Replace y with f-1x.00DefnsA function is a relation (a set of ordered pairs) where each x-value corresponds to exactly ONE y-value. A one-to-one (1-1) function is a function where each y-value corresponds to exactly ONE x-value.If f is a function, the inverse of f is a relation consisting of all ordered pairs y,x where x,y∈f. If f is 1-1, then its inverse is a function. The inverse function, denoted f-1, is defined asf-1=y,xx,y∈fWarning! f-1≠1fHorizontal Line Test (HLT)A function is one-to-one if every horizontal line interests the graph of the function at most once.4 Biggies of Inverse Functions'x' and 'y' values switch placesinverses reflect in the line y=x ff-1x=x and f-1fx=x The inverse is a function iff fx is one-to-one. How to Find the Inverse of a 1-1 FunctionStep 1: Replace fx with yStep 2: x?y (interchange x and y)Step 3: Solve for y.Step 4: Replace y with f-1x.Ex 1Find the inverse of fx=x2. What can we conclude?Consider fx=sinx. Define an inverse for sine.-182880113030Inverse Sine Functiony=sin-1x or y=arcsinx means that x=siny, for -π/2≤ y≤π/2.00Inverse Sine Functiony=sin-1x or y=arcsinx means that x=siny, for -π/2≤ y≤π/2.Note: The argument of inverse functions are real numbers and their output are angles. In particular, arcsine takes in a real number but ‘spits out’ an angle.Warning: sin-1x≠sinx-1That is, sin-1x≠1sinx. However, sin2x= ________=________________Notationarcsin12=θ means find _______ (ONE only) such that ______________________.vssinθ=12which means_____________________________________. This has _____________ solutions.Note: _________ is the ONLY correct solution to arcsin12.Why call it arcsin? Because θ (in radians) is the ____________________________________________________.Analogy to Elementary Algebra:Cosine is not 1-1 either (on its natural domain). Define an inverse for cosine.Inverse Tangent:-2921015875Inverse Cosine Functiony=cos-1x or y=arccosx means that x=cosy, for -0≤ y≤π.Inverse Tangent Functiony=tan-1x or y=arctanx means that x=tany, for -π/2< y<π/2.Inverse Cotangent, Secant, and Cosecant Functionsy=cot-1x or y=arccotx means that x=coty, for -0< y<π.y=sec-1x or y=arcsecx means that x=secy, for -0≤ y≤π, y≠π/2.y=csc-1x or y=arccscx means that x=cscy, for -π/2≤ y≤π/2, y≠0.00Inverse Cosine Functiony=cos-1x or y=arccosx means that x=cosy, for -0≤ y≤π.Inverse Tangent Functiony=tan-1x or y=arctanx means that x=tany, for -π/2< y<π/2.Inverse Cotangent, Secant, and Cosecant Functionsy=cot-1x or y=arccotx means that x=coty, for -0< y<π.y=sec-1x or y=arcsecx means that x=secy, for -0≤ y≤π, y≠π/2.y=csc-1x or y=arccscx means that x=cscy, for -π/2≤ y≤π/2, y≠0.Note:The inverse secant and inverse cosecant functions are sometimes defined with different ranges. We use intervals that match those of the inverse cosine and inverse sine functions (except for the missing point).InverseFunctionDomainRangeIntervalQuadrantsy=sin-1x-1,1-π2,π2y=cos-1x-1,10,πy=tan-1x-∞,∞-π2,π2y=cot-1x-∞,∞0,πy=sec-1x-∞,-1∪1,∞0,π2∪π2,πy=csc-1x-∞,-1∪1,∞-π2, 0∪0, π2Ex 2 (#8’)Consider the inverse cosine function, defined by y=cos-1x, or y=arccosx. a)Is this function increasing or decreasing?b)arccos-12=2π3. Why is arccos-12 not equal to -4π3?Ex 3 (#11)Is sec-1a calculated as cos-11a or as 1cos-1a?Ex 4Evaluate each if possible.(#14-36 evens) y=sin-1-1 (b) y=arctan-1c cos-1-12=(d) arccot-3= (e) sec-1-2= (f) arccsc-12= (g) arcsin-32= (h) arccsc-2= Ex 5Evaluate if possible. State answer in degrees.(#38)(#48) tan-13 b cos-1-2Ex 6Evaluate. a sinsin-132 b sin-1sinπ6 c coscos-132 d cos-1cosπ3(#80)(#84)e sinarccos14 f cos2sin-114(#86)(g) tan2cos-114 (h) cos-1cos-5π3 (#92) cossin-135+cos-1513 (j) cos-1sin7π6= Ex 7 Write as an algebraic (non-trigonometric) expression in u, u>0.(#102)cotarcsinu(#106)seccos-1uu2+5Sec 6.2: Trigonometric Equations I Ex 1Solve each equation for all x such that x∈0,2π.(#13)(#22)2sinx+3=4 2cos2x-cosx=1Ex 2 Solve each equation for θ such that θ∈0,360°. Note: On exam, numbers will work out nice (no calculators)Ex 3Solve each equation (x in radians and θ in degrees) for all exact values. Note: HW says round. Obtain exact answer first (this is your quiz/exam answer) then round (this is your hw answer). Write answers using the least possible nonnegative angle measures. Round degrees to the nearest tenth and radians to four decimal places. (#44)a tanθ+1=0 (b) (#50)c 4cos2x-1=0 (d) (e)Ex 4Grade the solution.Problem: Solve for all values of x∈0,2π that satisfy cos2x=1.Solution:cos2x2=12 ? cosx=12 ? x=π3Ex 5Solve for all values of x∈0,2π.a)6cotxcosx-3cotx=0b)c)tanθ+3=0 2sin2θ-3sinθ+1=0Factoring & Zero Factor Property ReviewName :_______________________ Date :_________________Factor Completely:1. 2. 3. 4. 5.6. 7. 8. 9. 10. 11. 12. 13. 14. Solve:Remember, if the equations are quadratic in form you have the following 4 tools:Quadratic FormulaFactoring (ZFP)Complete the SquareSquare Root Property – Don’t forget the ___________15. 16. 17. 18 Sec 6.3: Trigonometric Equations II Ex 1Solve each equation for x such that x?0,2π and for θ such that θ?0,360°.(#16)(#18)tan4x=0 cos2x-cosx=0(#20)(#24)sin2x2-2=0 sinxcosx=14Ex 2Solve each equation (x in radians and θ in degrees) for all exact values. Note: HW says round. Obtain exact answer first (this is your quiz/exam answer) then round (this is your hw answer). Write answers using the least possible nonnegative angle measures.(#34)(#36)sin2x=2cos2x cosx=sin2x2(#38’) Solve for only values in 0,2π(#42) Solve for only values in 0,2π4cos2x=8sinxcosx sinx2+cosx2=1Sec 6.4: Equations Involving Inverse Trigonometric Functions Ex 1Solve.(#12)(#16)y=-sinx3, for x∈-3π2, 3π2 y=tan2x-1, for x∈12-π4,12+π4(#20)(#28)y=4+3cosx, x∈0, π 4π+4tan-1x=π(#30)(#34)arccosx-π3=π6 cot-1x=tan-143(#38)(#40)arccosx+2arcsin32=π3 arcsin2x+arcsinx=π2(#42) PPsin-1x+tan-1x=0Ans: 0Solving Trigonometric EquationsI. Solve each equation for if .1) 2) 3) 4) II. Solve each equation for if .5) 6) 7) 8) 9) 10) Key Functions: LogarithmThe graph of is shown below. Use this graph to draw the graph of .*Label five ordered pairs on both graphs. *Use interval notation to list the following for the logarithmic function:Domain:_________________________Range:___________________________*What is the asymptote of the graph of? ___________________*What is the base in ? __________What is it called? _______________________*What is the base in ? _____________What is it called? _________________________1. Sketch the graph of each function. Draw and label the asymptote.a)b)c)d)2. Rewrite each logarithm in exponent form.a)b)3. Rewrite each exponent in logarithm form.a) b) 4. Solve each equation for all values of ‘x’ where .a)b)5. Evaluate where:a)b)6. Use log properties to rewrite. Simplify if possible.For any positive numbers M, N, a (), and any real number p: a) b) c) d) 7. Solve.a) b) Sec 7.4: Vectors, Operations, and Dot Product-19748554280DefnA vector is a physical quantity that has both magnitude and direction. Typical vector quantities include velocity, acceleration, and force. We use a directed line segment to represent a vector quantity. The length of the vector represents the magnitude and the direction of the vector, indicated by the arrowhead, represents the direction of the quantity.0DefnA vector is a physical quantity that has both magnitude and direction. Typical vector quantities include velocity, acceleration, and force. We use a directed line segment to represent a vector quantity. The length of the vector represents the magnitude and the direction of the vector, indicated by the arrowhead, represents the direction of the quantity.933456477000Vector Notation and FactsOPOPO is the initial point whereas P is the terminal pointOP≠POThe magnitude of the vector OP is OP. OP=POTwo vectors are equal iff (if and only if) the have the same direction and same magnitude.Same Vectors190506985-15862308953500215646034290The sum of two vectors is also a vector. We use the parallelogram rule to find the sum of two (geometric) vectors. Add “tail to tip”. Add vector A with vector B. The sum A+B is called the resultant vector.Is vector addition commutative?YESNOFor every vector v, there is a vector -v that has the same magnitude as v but opposite direction. Draw v, -v, 2v,u-v, and v-v.Ex 1Use the vectors in the figure to graph the vector.442569641326uvwuvw(#14) 2u-3v+w45783592075Ex:-16764041275DefnA vector with its initial point at the origin is called a position vector. A position vector u with its endpoint at the point a, b is written a. b. So, u=a,b.a=horizontal component and b= vertical componentGeometrically, a vector is a directed line segment whereas algebraically, it is like an ordered pair (whose initial point is at the origin).The positive angle formed between the positive x-axis and a position vector is called the direction angle.0DefnA vector with its initial point at the origin is called a position vector. A position vector u with its endpoint at the point a, b is written a. b. So, u=a,b.a=horizontal component and b= vertical componentGeometrically, a vector is a directed line segment whereas algebraically, it is like an ordered pair (whose initial point is at the origin).The positive angle formed between the positive x-axis and a position vector is called the direction angle.56388115875Magnitude and Direction Angle of a Vector a,bThe magnitude (length) of vector u=a,b is given by the following.u=a2+b2The direction angle θ satisfies tanθ=ba, a≠000Magnitude and Direction Angle of a Vector a,bThe magnitude (length) of vector u=a,b is given by the following.u=a2+b2The direction angle θ satisfies tanθ=ba, a≠0-1460578105Horizontal and Vertical ComponentsThe horizontal and vertical components, respectively, of a vector u having magnitude u and direction angle θ are the following.a=ucos θ and b=usin θThat is, u=a,b=ucos θ,usin θHorizontal and Vertical ComponentsThe horizontal and vertical components, respectively, of a vector u having magnitude u and direction angle θ are the following.a=ucos θ and b=usin θThat is, u=a,b=ucos θ,usin θ-14605113665Parallelogram PropertiesA parallelogram is a quadrilateral whose opposite sides are parallel.The opposite sides and opposite angles of a parallelogram are equal, and adjacent angles of a parallelogram are supplementary.The diagonals of a parallelogram bisect each other, but they do not necessarily bisect the angles of the parallelogram.Parallelogram PropertiesA parallelogram is a quadrilateral whose opposite sides are parallel.The opposite sides and opposite angles of a parallelogram are equal, and adjacent angles of a parallelogram are supplementary.The diagonals of a parallelogram bisect each other, but they do not necessarily bisect the angles of the parallelogram.4502156350Vector OperationsLet a, b, c,d, and k represent real numbers.a,b+c,d=a+c,b+dk?a,b=ka,kb (k is a called a scalar)If u=a,b, then –u=-a,-ba,b-c,d=a,b+-c,d=a-c,b-d0Vector OperationsLet a, b, c,d, and k represent real numbers.a,b+c,d=a+c,b+dk?a,b=ka,kb (k is a called a scalar)If u=a,b, then –u=-a,-ba,b-c,d=a,b+-c,d=a-c,b-d-9525031115DefnA unit vector is a vector that has magnitude 1. Two useful vectors are i=i=1,0 and j=j=0,1 and are graphed below.If v=a,b, then v=ai+bj.DefnA unit vector is a vector that has magnitude 1. Two useful vectors are i=i=1,0 and j=j=0,1 and are graphed below.If v=a,b, then v=ai+bj.Ex 2Given vectors u and v, find (a) 2u(b) 2u+3v(c) v-3u(#26)(#28)u=-i+2j, v=i-j u=-2, -1, v=-3, 2The dot product of two vectors is a scalar (a real number), NOT a vector. It is also known as the inner product. Dot products are used to determine the angle between two vectors, to derive geometric theorems, and to solve physical problems. (Once we have the formula for the geometric interpretation of the dot product, we will see that we can think of the dot product u?v as how much the projection of u is going in the same direction of v.)-9461546685Dot ProductThe dot product of the two vectors u=a,b and v=c,d is denoted ?v , read “u dot v” and is given by the following.u?v=ac+bdNote:The dot product is a scalar, not a vector!0Dot ProductThe dot product of the two vectors u=a,b and v=c,d is denoted ?v , read “u dot v” and is given by the following.u?v=ac+bdNote:The dot product is a scalar, not a vector!291465-235585Properties of the Dot ProductFor all vectors u, v, and w and scalars k, the following hold.(a)u?v=v?u(b)u?v+w=u?v+u?w(c)u+v?w=u?w+v?w(d)ku?v=ku?v=u?kv(e)0?u=0(f)u?u=u2Properties of the Dot ProductFor all vectors u, v, and w and scalars k, the following hold.(a)u?v=v?u(b)u?v+w=u?v+u?w(c)u+v?w=u?w+v?w(d)ku?v=ku?v=u?kv(e)0?u=0(f)u?u=u2Derive the Formula for the Geometric Interpretation of the Dot Product (for 2-dim) (using Law Cosines)-182880114173Geometric Interpretation of the Dot ProductIf θ is the angle between the two nonzero vectors u and v, where 0°≤θ≤180°, then the following holds.cosθ=u?vuvNote:If u?v>0, the angle is acute.If u?v=0, the angle is a right angle. (The vectors are thus perpendicular or orthogonal.)If u?v<0, the angle is obtuse.00Geometric Interpretation of the Dot ProductIf θ is the angle between the two nonzero vectors u and v, where 0°≤θ≤180°, then the following holds.cosθ=u?vuvNote:If u?v>0, the angle is acute.If u?v=0, the angle is a right angle. (The vectors are thus perpendicular or orthogonal.)If u?v<0, the angle is obtuse.Ex 3 (#30)For the pair of vectors with angle θ between them, sketch the resultant.u=8, v=12, θ=20°Ex 4 (#34)Find the magnitude and direction angle for the vector -7, 24.Ex 5 (#42)Find the magnitude of the horizontal and vertical components of v, if θ is the direction angle.θ=146.3°, v=238Ex 6Write each vector in the form a,b.(#44)(#48)2926169164Ex 7 (#52)Two forces of 37.8 lb and 53.7 lb act at a point in the plane. The angle between the two forces is 68.5°. Find the magnitude of the resultant force.Ex 8 (#56)Find the magnitude of the resultant force using the parallelogram rule.-3683013017500Ex 9 (#66)Given u=-2,5 and v=4,3, find 2u+v-6v.Ex 10Find the dot product for each pair of vectors.(#74)(#76)7,-2, 4,142i+4j, -jEx 11 (#78)Find the angle between 1,7 and 1,1.Ex 12Determine whether each pair of vectors is orthogonal.(#88)(#90)(#92)1,1, 1,-1 3,4, 6,8-4i+3j, 8i-6jSection 7-5 Application of Vectors (lecture examples)Read each problem carefully and start with a sketch of the situation. Then use what you know about vectors to solve each problem. 1) An arrow is shot into the air so that its horizontal velocity is 25 feet per second and its vertical velocity is 15 feet per second. Find the velocity of the arrow.2) A boat is crossing a river that runs due north. The heading of the boat is due east, and it is moving through the water at 12.0 mph. If the current of the river is a constant 3.25 mph, find the true course of the boat.3) An airplane has a velocity of 400mph southwest. A 50mph wind is blowing from the west. Find the resultant speed AND direction of the plane.4) A plane is flying at 170 miles per hour with heading 52.5 due north. The wind currents are a constant 35.0 miles per hour at 142.5 due north. Find the ground speed and true course of the plane.5) Two rescue vessels are pulling a broken-down motorboat toward a boathouse with forces of 840 lb and 960 lb. The angle between these forces is 24.5. Find the magnitude and the direction of the equilibrant force with the 840 lb force.6) Find the force required to keep a 2000-lb car parked on a hill that makes and angle of 30 with the horizontal.Sec 7.5: Applications of VectorsEx 1 (#8)Find the force required to keep a 3000-lb car parked on a hill that makes an angle of 15° with the horizontal. Ex 2 (#21)An airline route from San Francisco to Honolulu is on a bearing of 233.0°. A jet flying at 450 mph on that bearing encounters a wind blowing at 39.0 mph from direction of a 114.0°. Find the resulting bearing and ground speed of the plane.Ex 3 (#25)A pilot is flying at 190.0 mph. He wants his flight path to be on a bearing of 64°30'. A wind is blowing from the south at 35.0 mph. Find the bearing he should fly, and find the plane’s ground speed.Ex 4 (#29)An airplane is headed on a bearing of 174° at an airspeed of 240 km per hr. A 30-km-per-hr wind is blowing from a direction of 245°. Find the ground speed and resulting bearing of the plane.Sec 8.1: Complex Numbers-35113026289DefnsThe imaginary unit i is equal to -1. i=-1 and therefore i2=-1A complex number has the form a+bi where a and b are real numbers. a is the real part whereas b is the imaginary part.Arithmetic with Complex NumbersFor complex numbers a+bi and c+di, a+bi+c+di=a+c+b+dianda+bi-c+di=a-c+b-diWe add/subtract the real parts and add/subtract the imaginary parts.a+bic+di=ac-bd+ad+bciProperty of Complex ConjugatesFor real numbers a and b,a+bia-bi=a2+b2DefnsThe imaginary unit i is equal to -1. i=-1 and therefore i2=-1A complex number has the form a+bi where a and b are real numbers. a is the real part whereas b is the imaginary part.Arithmetic with Complex NumbersFor complex numbers a+bi and c+di, a+bi+c+di=a+c+b+dianda+bi-c+di=a-c+b-diWe add/subtract the real parts and add/subtract the imaginary parts.a+bic+di=ac-bd+ad+bciProperty of Complex ConjugatesFor real numbers a and b,a+bia-bi=a2+b2Cycle of iEx 1 Identify each number as rational (real), irrational (real), pure imaginary (complex), or nonreal complex. (More than one description may apply.)-43 00 e+13i -7i 5+i-6-2i π 24 -25 6--36Ex 2Solve.(#30)(#36)2x2+3x=-2x2+2=2xEx 3Simplify. Write answers in a+bi form (standard form) when possible.(#22)(#24)(#38)-500 --80 -17?-17(#40)(#48)(#52)-5?-15 -12?-68 20+-82(#56)(#62)(#74)4-i+8+5i 37-47-i-4i+-27+5i 2-4i2+4i(#78)(#80)(#84)-5i4-3i2 3-i3+i2-6i 4-3i4+3i(#92)(#102)(#104)59i i-14 1i-12Ex 4 (#107)Show that α is a square root of i.α=22+22iSec 8.2: Trigonometric (Polar) Form of Complex NumbersWe cannot order the complex numbers (as we can with R) but ? a 1-1 correspondence between the set of all complex numbers, C, and the set of all 2-dimensional vectors (2x1 matrices), R2.4871720154305Plot several complex numbers.Plot a few complex numbers and add them. What is the relationship between the sum of complex numbers and the sum of vectors?What is the rectangular form of a complex number? Establish relationships among x, y, r, and θ using the figure.298378979502Relationships Among x, y, r, and θ1)2)4)3)0Relationships Among x, y, r, and θ1)2)4)3)-6583748082x+yix+yiWhat is the polar form of a complex number?1270116205Polar Form of a Complex NumberThe polar (trigonometric) form of the complex number z=x+yi is z=rcosθ+isinθwhere r=z and θ is the angle direction angle of the vector x,y. r is called the modulus of z and θ is called the argument of z. We often choose θ∈0, 360°. Abbreviation: rcosθ+isinθ=r cis θ00Polar Form of a Complex NumberThe polar (trigonometric) form of the complex number z=x+yi is z=rcosθ+isinθwhere r=z and θ is the angle direction angle of the vector x,y. r is called the modulus of z and θ is called the argument of z. We often choose θ∈0, 360°. Abbreviation: rcosθ+isinθ=r cis θEx 1Find the sum of the complex numbers. Graph each and their resultant.(#24)2+3i and-4+2i -15+27i and37-34iEx 2Write each complex number in rectangular form.(#34)(#38)6 cis 135° 2cos-60°+isin-60°Ex 3Write each complex number in polar form with θ∈0, 360°. (#40)(#44)(#48)1+i3 -2+2i -2iEx 4 (#59-62)Describe the graphs of all complex numbers z satisfying the specified conditions.The absolute value of z is 1.The real and imaginary parts of z are equal.The real part of z is 1.The imaginary part of z is 1.Sec 8.3: The Product and Quotient TheoremsFind z1z2 where z1=r1cisθ1 and z2=r2cisθ2.Find where z1=r1cisθ1 and z2=r2cisθ2.-109220161290Product TheoremIf r1cosθ1+isinθ1 and r2cosθ2+isinθ2 are any two complex numbers, then the following holds.r1cosθ1+isinθ1?r2cosθ2+isinθ2=r1r2cosθ1+θ2+isinθ1+θ2In compact form:? r1cisθ1r2cisθ2=r1r2cis θ1+θ2Multiply the lengths. Add the angles.Quotient TheoremIf r2cosθ2+isinθ2≠0, then the following holds.r1cosθ1+isinθ1r2cosθ2+isinθ2=r1r2cosθ1-θ2+isinθ1-θ2In compact form:? r1cisθ1r2cisθ2=r1r2cis θ1-θ2Divide the lengths. Subtract the angles.0Product TheoremIf r1cosθ1+isinθ1 and r2cosθ2+isinθ2 are any two complex numbers, then the following holds.r1cosθ1+isinθ1?r2cosθ2+isinθ2=r1r2cosθ1+θ2+isinθ1+θ2In compact form:? r1cisθ1r2cisθ2=r1r2cis θ1+θ2Multiply the lengths. Add the angles.Quotient TheoremIf r2cosθ2+isinθ2≠0, then the following holds.r1cosθ1+isinθ1r2cosθ2+isinθ2=r1r2cosθ1-θ2+isinθ1-θ2In compact form:? r1cisθ1r2cisθ2=r1r2cis θ1-θ2Divide the lengths. Subtract the angles.Ex 1Find each and write in rectangular form.(#8)(#12)8cos210°+isin210°2cos330°+isin330° 3cis300°7cis270°(#16)12cos23°+isin23°6cos293°+isin293°(#20) (#24)2i-1-i3 -32+3i66+i2Sec 8.4: De Moivre’s Theorem; Powers and Roots of Complex Numbers-7315262078De Moivre’s TheoremIf z=rcosθ+isinθ is a complex number and if n is any natural number then the following holds.rcosθ+isinθn=rncosnθ+isinnθIn compact form:r cis θn=rncis nθDefnFor a natural number n, the complex number a+bi is an nth root of the complex number x+yi ifa+bin=x+yinth Root TheoremIf n is natural number, r is a positive real number, and θ is in degrees, then the nonzero complex number rcosθ+isinθ has exactly n distinct nth roots given by the following.nrcosα+isinα or nr cis αwhere α=θ+360°kn or α=θn+360°kn , k=0, 1, 2, …, n-1If θ is in radians, then α=θ+2πkn or α=θn+2πkn , k=0, 1, 2, …, n-1De Moivre’s TheoremIf z=rcosθ+isinθ is a complex number and if n is any natural number then the following holds.rcosθ+isinθn=rncosnθ+isinnθIn compact form:r cis θn=rncis nθDefnFor a natural number n, the complex number a+bi is an nth root of the complex number x+yi ifa+bin=x+yinth Root TheoremIf n is natural number, r is a positive real number, and θ is in degrees, then the nonzero complex number rcosθ+isinθ has exactly n distinct nth roots given by the following.nrcosα+isinα or nr cis αwhere α=θ+360°kn or α=θn+360°kn , k=0, 1, 2, …, n-1If θ is in radians, then α=θ+2πkn or α=θn+2πkn , k=0, 1, 2, …, n-1Proof of nth Root Theorem:Ex 1Find each power. Write in rectangular form.(#4) PP(#6)2cos120°+isin120°3 3cis40°3Ans: 8(#10)22-22i8Ex 2(a) Find all cube roots of each complex number. Leave answers in polar form.(b) Graph each cube root as a vector in the complex plane.(#20)27 Ex 3 (#30)Find and graph all fourth roots of i.Ex 4 Solve for all (complex) solutions. Leave answer in rectangular form.x3+1=0Try problems:Sec 8.5: Polar Equations and GraphsWhy polar coordinates?Different, sometimes more efficient, way to represent a point or graph. (e.g. shortest distance)Makes some applications/equations easier to work with (e.g. r=3 vs ____________________ and using DeMoivre's Theorem vs Binomial Theorem)Some non-functions in rectangular coordinates are functions in polar coordinates so we can apply rules/theorems of functions that we wouldn't be able to otherwiseRectangular Coordinates VS Polar Coordinates27051010223500403799020676PolePolar AxisCoordinates of the pole ___________Examples: 4, 270°1,π6-3, 120°-2, -π2-2070102349500Are points uniquely defined? -8890158115Polar Coordinates Rectangular Coordinatesr, θ?x,yx=rcosθ y=rsinθ r2=x2+y2 tanθ=yx, x≠000Polar Coordinates Rectangular Coordinatesr, θ?x,yx=rcosθ y=rsinθ r2=x2+y2 tanθ=yx, x≠0Ex 1 Find the rectangular coordinates of each point.-3,π -2,2π3 -2, -π4 Ex 2 Find the polar coordinates of each rectangular coordinate. 0,-2 -3, 3-2, -23 Ex 3 Rewrite each rectangular equation in polar equation.x2+y2=x x2=4y x=4 y=-3Ex 4 Rewrite each polar equation in rectangular equation.r=sinθ r=2 r=33-cosθcenter4445Symmetry WRT…Polar axis Replace θ by -θ.Line θ=π2 Replace θ by π-θPole Replace r by -r00Symmetry WRT…Polar axis Replace θ by -θ.Line θ=π2 Replace θ by π-θPole Replace r by -rEx 5 Transform each polar equation into rectangular equation. Then identify and graph the equation. r=2 θ=-π4 r=2sinθ-979931668023248660-3810908050-3810Lines Should be able to convert to polar form. Need to know the graphs of polar form for all but y=mx+b type.Polar FormGraphy=5y=-2x=4x=-3y=mx (y=2x)y=mx+b----------------------------------------------------ax+by=cCircles Rectangular FormGraphr=3r=2sinθRose Curves r=4cos3θr=4sin3θ3 petalsr=4cos2θr=4sin2θ4 petalsr=-4cos3θr=-4sin3θ3 petalsr=-4cos2θr=-4sin2θ4 petals()-2921027305r=5cos3θr=5sin3θr=5cos2θr=5sin2θr=5cos5θr=5sin5θr=5cos4θr=5sin4θr=5cos7θr=5sin7θr=5cos6θr=5sin6θ00r=5cos3θr=5sin3θr=5cos2θr=5sin2θr=5cos5θr=5sin5θr=5cos4θr=5sin4θr=5cos7θr=5sin7θr=5cos6θr=5sin6θNOTE:Cosine rose curves start on _________________ and Sine rose curves start in __________ and has a petal on the __________________-__________.CardioidPasses through poler=2+2cosθr=2-2cosθr=2+2sinθr=2-2sinθLima?on w/out Inner LoopDoes NOT pass through pole r=3+2cosθr=3-2cosθr=3+2sinθr=3-2sinθLima?on w/Inner LoopPasses through poler=1+2cosθr=1-2cosθr=1+2sinθr=1-2sinθLemniscateFigure 8r2=4sin2θr2=4cos2θ-31757048500-31754561840-107951825625-10795-186055The Cycle of a Rose CurveFor each Rose Curve, fill in enough of the table to complete the graph one time.1)0r2)0r3)0r4)0rName :____________________________Polar Equations and Graphs1. 2. 3. 4. 5. 6. 7. 8. Math 335 Polar Equations and Graphs Homework HandoutUse pencil and circle final answers.Convert each rectangular equation into polar equation. (solve for ‘r =’ whenever appropriate)1)2)3)4)5)6)7)8)Convert each polar equation into rectangular equation.9)10)11)12)13)14)Problems 15 – 24. Graph each polar equation.15)16) 17)r = 218)19)20)21)22)23)24)Problems 25 – 26. Graph each of the polar equations.25)0r26)0r27. Circle the graph that matches the given polar equation.(a)(b) (c)(d)28.a.b.c.d.29. a.b.c.d.30. a.b.c.d.Graph each polar equations.31. 32. 2051051752600033. 34. 128905654050035. 36. ................
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