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Lesson 13.1 & 13.2: Right Triangle Trigonometry &Reciprocal Trig FunctionsLearning Goals:What are the three trigonometry ratios? What are the purposes of these ratios?How do we find a missing side or angle of a right triangle?What are the reciprocal trig functions?How do we create a right triangle when given a trig ratio?Do Now: In each triangle, label the appropriate sides as adjacent, opposite, and hypotenuse, with respect to the marked acute angle.What are the 3 trigonometry ratios?S OH C AHT OAsinθ=opphyp cosθ=adjhyptanθ=oppadjWhat does θ represent? θ= angle in referenceWhat are the purposes of these ratios? Find a missing side or angle of right triangles. a) What are the values of b) What are the values of sinA, cosA, andtanA?sinB, cosB, andtanB?sinA=1213, cosA=513, tanA=125 sinB=513, cosB=1213, tanB=512 What does the term “reciprocal” mean? To flip a function, i.e. 23→32This must be memorized:Cosecant (csc) is the reciprocal trigonometry function of sinecsc θ=1 sinθSecant (sec) is the reciprocal trigonometry function of cosinesecθ=1cosθCotangent (cot) is the reciprocal trigonometry function of tangentcotθ=1tanθ449580041148000Example: Find the values of the six trigonometric functions for angle A. First find the missing side BC using Pythagorean Theorem. a2+b2=c2 sinA=513 cscA=135a2+122=132 cosA=1213 secA=1312a2=25 tanA=512 cotA=125a=5 Finding a Side4791075327025001. Find the length, to the nearest tenth, of the leg adjacent to the 40° angle.What ratio? C AH=cosθ=adjhypcos40=x18 Cross Multiply! (make sure you are in degree mode)x=18(cos40)=13.8 2. Find the length, to the nearest hundredth, of the leg opposite the 60° angle. What ratio? T OA=tanθ=oppadj tan60=x.7 Cross Multiply! (make sure you are in degree mode)x=.7(tan60)=1.21 3. Find the length of the hypotenuse, to the nearest whole number.What ratio? S OH=sinθ=opphyp sin45=7.1x Cross Multiply! (make sure you are in degree mode)xsin45=7.1 xsin45sin45=7.1sin45 x=10 Finding an Angle4398453517525004. A 20-foot ladder leaning against a vertical wall reaches to a height of 16 feet. Find the value of the angle, to the nearest degree, that the ladder makes with the ground. Draw a picture!Which ratio? S OH=sinθ=opphyp sinx=1620 inverse to get the angle! 2nd→sinx=sin-11620=53° 4270375533400005*. In right triangle ABC shown below, AB=18.3 and BC=11.2. What is the measure of angle C, to the nearest tenth of a degree? Which ratio? T OA=tanθ=oppadj tanC=18.311.2 inverse to get the angle! 2nd→tanC=tan-118.311.2=58.5° 4486275499745006. The height of a vertical cliff is 450 m. The angle of elevation from a ship to the top of the cliff is 23°. The ship is x meters from the bottom of the cliff.a) Draw a diagram to show this information.b) Calculate the value of x.Which ratio? T OA=tanθ=oppadj tan23=450x Cross Multiply! (make sure you are in degree mode)x tan23=450 xtan23tan23=450tan23 x=1060.134 3676650958215007. Standing on the gallery of a lighthouse (the deck at the top of a lighthouse), a person spots a ship at an angle of depression of 20°. The lighthouse is 28 m tall and sits on a cliff 45 m tall as measured from sea level. What is the horizontal distance, to the nearest tenth between the lighthouse and the ship? Sketch and label a diagram to support your answer.Which ratio? T OA=tanθ=oppadj tan20=73x Cross Multiply! (make sure you are in degree mode)x tan20=73 xtan20tan20=73tan20 x=1200.6 8. In triangle ABC, C is the right angle, AB=4, and CB=3. Evaluate the six trigonometric ratios of angle B and write all answers in simplest radical form.519303015875000Draw a diagram!First find the missing side using Pythagorean Theorem. a2+b2=c2 sinB=74 cscB=47?77=47732+b2=42 cosB=34 secB=43b2=7 tanB=73 cotB=37?77=377b=7 4419600-194310009. Find the value of the trig function indicated.a) Find cscθ if tanθ=512=oppadjsinθ=513 cscθ=135 4810125000b) Find cosθ if cscθ=1.75convert to a fraction(ratio) 1.75→math→frac=74cscθ=74 sinθ=47cosθ=337 -32385015430500c) Find tanθ if secθ=2secθ=21=hypadjtanθ=31=3 5010150000d) Find secθ if sinθ=31313=opphypsecθ=1352 ?5252=135252=524=4?134=2134=132 Homework 13.1 & 13.2: Right Triangle Trigonometry & Reciprocal Trig Functions3800475480695001. A tree casts a 25-foot shadow on a sunny day, as shown in the diagram below. If the angle of elevation from the tip of the shadow to the top of the tree is 32°, what is the height of the tree to the nearest tenth of a foot?2. A 20-foot ladder leaning against a vertical wall reaches to a height of 16 feet. Find the value of the angle, to the nearest degree, that the ladder makes with the ground.3. A tree casts a shadow that is 20 feet long. The angle of elevation from the end of the shadow to the top of the tree is 66°. Determine the height of the tree, to the nearest foot.4. A ship on the ocean surface detects a sunken ship on the ocean floor at an angle of depression of 50°. The distance between the ship on the surface and the sunken ship on the ocean floor is 200 meters. If the ocean floor is level in this area, how far above the ocean floor, to the nearest meter, is the ship on the surface?4276725306705005. In the diagram of right triangle ABC shown below, AB=14 and AC=9. What is the measure of ∠A, to the nearest degree?(1) 33 (2) 40 (3) 50 (4) 57426720052705006. Which ratio represents cscA in the diagram below?(1) 2524 (2) 257 (3) 247 (4) 7247. Find cosθ, if cscθ=1358. Find cotθ if secθ=1.259. In the right triangle shown below, what is the measure of angle S, to the nearest tenth of a degree?45233003333750010. In the right triangle shown in the diagram below, what is the value of x to the nearest whole number? Lesson 13.3: Angles of RotationLearning Goals:How do we graph and label angles in standard position?What are the coterminal angles? How do we find coterminal angles?479107511176000Initial Side – the ray (side) at which an angle of rotation begins … always positive x-axis.Terminal Side – the ray (side) at which an angle of rotation ends.Standard Position – an angle is in standard position if its vertex is located at the origin and one ray is on the positive x-axis.Practice: Draw and label an angle with the given measure in standard position.If the angle is negative, you can keep adding 360° until you have a positive angle.If the angle is larger than 360°, you can keep subtracting 360° until the angle is between 0° and 360°.1. 150°2. -30°3. -220° 4. 300°5. 270°6. 400° 7. -550°8. 965° Coterminal AnglesAny angle that share the same terminal side (add/subtract 360°)Graph and label 120° and -240° on the same graph. What do you notice about the terminal side of each angle? 1. Find a negative degree measure that is co-terminal with 320°.320-360=-40° 2. Find a positive degree measure that is co-terminal with 320°.320+360=680° 3. Find the degree measures of 2 angles, one positive and one negative, that are co-terminal with -500°.-500+360=-140° -140+360=220° 48101258572500RadiansWhat is a radian? A radian is the measure of an angle that, when drawn as a central angle of a circle, intercepts an arc whose length is equal to the length of a radius of the circle. Radians is just another way to measure an angle!What is the measure of π radians in degree?π=180° π=3.14both are radians!How to convert between Degrees and RadiansMethod 1: ProportionDR=180πMethod 2: On Reference Sheet1 radian=180π degrees convert to degrees1 degree=π180 radians convert to radiansExamples: Convert the following to radian measure.a) 50°=501?π180=50π180=5π18b) -120°=-1201?π180=-120π180=-2π3Examples: Convert the following to degree measure. If π is given just substitute π=180!a) π6=1806=30°b) -2π5=-21805=-3605=-72°1. Find, to the nearest degree, the angle whose measure is 3.45 radians.D3.45=180π or3.45?180π=621π=198° π?D=621 π?Dπ=621π D=198° 2. What is the radian measure, in terms of π, of the angle formed by the hands of a clock at 4:00 p.m.?36012=30?4=120°?π180=120π180=2π3 3. How many radians does the minute hand of a clock rotate through over 10 minutes? How many degrees?36012=30?2=60°?π180=60π180=π3 379095048895004. Sketch and label θ in standard position if θ=7π6=7π6?180π=12606=210°..Homework 13.3: Angles of Rotation1. Which angle is coterminal with an angle of 125°?(1) -125° (2) -235° (3) 235° (4) 425°2. Find the degree measures of 2 angles, one positive and one negative, that are coterminal with 45°.3. Draw and label an angle with the given measure in standard position.a. 260°b. 180°c. -120° 4867275198120004. In what quadrant does the terminal side of angle 520° lie?(1) I (2) II (3) III (4) IV5. Find the measure, in radians, of the smaller angle formed by the hands of a clock at the following time:6. What is 235°, expressed in radian measure in terms of π?7. What is the number of degrees in an angle whose radian measure is 7π12?8. Find, to the nearest tenth of a degree, the angle whose measure is 2.75 radians.Lesson 13.4: Special Right Triangles and Exact ValueLeft-Hand RuleLearning Goals:What are the special right triangles and how do we use them to find the exact value of the trigonometric functions for angles in the first quadrant?What is the co-function relationship and how do we use it?Special Right TrianglesTo determine the sine, cosine, and tangent values at 30°, 45°, and 60°, there are 2 special right triangles that we can get the values from.Below are the triangles where these trig values come from.Warm-Up: Use the triangles above and what you already know about right triangles, to find the trig values listed in the chart below. Be sure to rationalize all denominators.0°30°45°60°90°sinθ01222321cosθ13222120tanθ013=3313undefinedDetermine the exact value of each of the following expressions in simplest radical form:1. sin30°+cos30°12+32=1+32 2. (csc45°)(tan60°)1sin45=122=1?22=22?22=222=21sin45tan60=2?3=6 3. cos60°+sin60°-cot60°1tan60=13=13?33=3312+32-33 1(3)2(3)+3(3)2(3)-3(2)3(2) Common Denominator36+336-236 3+36 4. sec30°+tan30°1cos30=132=23=23?33=233233+33 333 3 5162550361950005. If sinθ=22 and θ is in quadrant I, find tanθ.θ=sin-122=45° tan45=1 6. If tanθ=33 and θ is in quadrant I, find cosθ.θ=tan-133=30° cos30=3 2 4533900000Let’s explore some other relationships…Answer the following questions with a partner.What type of triangle is shown here? Right How do you know? Right AngleUsing your answer from above and your knowledge of the interior angles of a triangle, what MUST be the sum of θ and φ? θ+φ=90° 1905046355Therefore the two non-right angles in a right triangle must always be complementary.0Therefore the two non-right angles in a right triangle must always be complementary.Write the indicated trig ratios using the sides of the triangle shown above:cosθ=45 sinφ=45a. What do you notice about these values? The ratios are the same & angles are complementary.b. Will this be true for ALL right triangles? Try another! ∠x+∠y=90°cosx=610 siny=610-104775349250The VALUE of sine of an acute angle is equal to the cosine of its complement.Symbols: sinθ=cos(90°-θ) example: sin25=cos65The VALUE of cosine of an acute angle is equal to the sine of its complement.Symbols: cosθ=sin(90°-θ)example: cos50=sin40VALUES are equal, ANGLES are complements!00The VALUE of sine of an acute angle is equal to the cosine of its complement.Symbols: sinθ=cos(90°-θ) example: sin25=cos65The VALUE of cosine of an acute angle is equal to the sine of its complement.Symbols: cosθ=sin(90°-θ)example: cos50=sin40VALUES are equal, ANGLES are complements!ratios are equalangles are complementary cofunctions 1. Write each expression as a function of an acute angle whose measure is less than 45°.a. sin80°=cos10°b. cos56°=sin34°2. Each equation contains the measures of two acute angles. Find a value of θ for which the statement is true.a. sin10°=cosθsinθ &cosθ are cofunctions, so they are complementary10+θ=90 θ=80° 3. In right triangle ABC with the right angle at C, sinA=2x+0.1 and cosB=4x-0.7. Determine and state the value of x. Explain your answer.They are cofunctions, so are complementary!2x+0.1+4x-0.7=90 6x-.6=90 6x=90.6 x=15.1 4*. Express sin390° in terms of its cofunction.sin390=cos(90-390)=cos(-300) 5*. If x is an acute angle and sinx=35, then (1) cosx=35 (2) cosx=25 (3) cos90-x=25 (4) cos90-x=356*. Suppose θ represents a number of degrees of rotation. Is it possible that cosθ°=12 and sinθ°=12? Explain.7*. Given that the cos90-x=45, find cotxHomework 13.4: Special Right Triangles and Exact Value1. Find the exact value of the following expressions:a. sin60°tan30°b. (sin45°)(cos45°)(tan45°)c. tan30°+cos30°2. If sinx+30=cos?(2x+80), find the value of x.3. Rewrite sin75° as a function of a positive angle less than 45°.4. A ladder leans against a house forming an angle of 60° between the ladder and the ground. The base of the ladder is 8 feet from the house.a. How long is the ladder? Write your answer in exact form?b. How high up the house does it reach? Write your answer in exact form.5. If sinθ=35, find (secθ)(cotθ) in simplest form.Lesson 13.5: The Unit CircleLearning Goals:What is the unit circle and how does it give us the values of the 6 trig functions?How do we determine the sign of each of the trig values in the four quadrants?Unit Circle TrigonometryThe Unit Circle is the circle centered at the origin with radius 1 unit (hence, the “unit” circle). The equation of this circle is x2+y2=1. A diagram of the unit circle is shown below:220980017780000319087562865000Exercise: A line segment is constructed from the origin, intersecting the circle at (x, y) in Quadrant I, as shown. The line segment makes an angle θ with the positive x-axis. Label the angle θ. Find an expression in terms of x and y for sinθ, cosθ, and tanθ.-9525195262500sinθ=y1=y cosθ=x1=x tanθ=yx or sinθcosθ x, y=(cosθ, sinθ) Practice with the Unit Circle:3667226597535001. Using the unit circle below, find an expression for the six trigonometric functions, in terms of x and y.sinθ=y1=y cscθ=1ycosθ=x1=x secθ=1xtanθ=yx or sinθcosθ cotθ=xy=cosθsinθ4019550228600002. The accompanying diagram shows unit circle O. Which line segments have a length of 1? Why? OC & OB they are a radius of the circlea. Which line segment has a length equivalent to sinθ?(1) OB (2) CD (3) OD (4) BAb. Which line segment has a length equivalent to cosθ?(1) OB (2) CD (3) OD (4) OAc. Which line segment has a length equivalent to tanθ?(1) OB (2) CD (3) OC (4) OAFinding Exact Values Not In The Unit Circle (Radius is NOT 1)3. Consider the following diagram of a circle of radius r centered at the origin. The line l is tangent to the circle at S (r, 0), so l is perpendicular to the x-axis.For the given values of r and θ, find tθ=30, r=3θ=45, r=4tan30=t3 tan45=t43tan30=t 4tan45=t t=3?33=3 t=4?1=4 4. Circle O has a radius of 2 units. An angle with a measure of 30° is in standard position. If the terminal side of the angle intersects the circle at point B, what are the coordinates of B?(1) 32,12 (2) 3,1 (3) 12,32 (4) 1,3304560y sinθ122232x cosθ32221232,12?2 (the radius)=232,22=3,1Quadrants of the Unit CircleWhen we include negative values, the x and y axes divide the space up into 4 pieces: Quadrant I, II, III, and IV. Recall: Earlier in this lesson, we discussed that a point on the unit circle can be rewritten using trig function.x, y=cosθ, sinθ tanθ=yx=sinθcosθ4191000276225004648200-16383000There is a pattern! Look at when Sine Cosine and Tangent are positive…All of them are positive in Quadrant ISine is positive in Quadrant IITangent is positive in Quadrant IIICosine is positive in Quadrant IV1. If sinA<0 and tanA>0, in which quadrant does the terminal side of ∠A lie?(1) I (2) II (3) III (4) IV2. If cosθ>0 and cscθ<0, in which quadrant does the terminal side of θ lie?(1) I (2) II (3) III (4) IV3*. If sinx=-13 and sinxcosx>0, in which quadrant does angle x lie?(1) I (2) II (3) III (4) IVHomework 13.5: The Unit Circle1. If sinθ=1-174, then angle θ lies in which quadrants?(1) I and II, only (2) II and IV, only (3) III and IV, only (4) I, II, III, and IV2. If cosx=-0.7 and cscx>0, the terminal side of angle x is located in Quadrant(1) I (2) II (3) III (4) IV47625363855003. Using the unit circle below, explain why secθ=1x4. State the sinθ, cosθ, and tanθ if the terminal side of the angle intersects the unit circle at (0.97, -0.26)3924300517525005. In the accompanying diagram, PR is tangent to circle O at R, QS ⊥ OR, and PR ⊥ OR. Which measure represents sinθ?(1) SO (2) RO (3) PR (4) QS398145039052506. Consider the following diagram of a circle of radius r centered at the origin. The line l is tangent to the circle at S (r, 0), so l is perpendicular to the x-axis.For the given values of r and θ, find t.a. θ=45, r=2b. θ=60, r=2Lesson 13.6: Reference Angles and Exact ValuesLearning Goals:What is a reference angle and how do we find the reference angle in each quadrant? Why are reference angles important?How do we determine the exact value of a trig expression?Do Now: Complete the following questions to prepare for today’s lesson.a) Sketch and label θ in standard position if θ=7π6? 035877500θ=7π6=71806=210 convert to degrees so it is easier to sketch!493395063500b) If sinθ<0 and tanθ>0, in what quadrant does the terminal side of θ lie?57245252413000Use ASTC, Q 3sinθ<0 means - and tanθ>0 means (+)What is a reference angle?Reference angle is the acute angle formed by the terminal side of the angle and the x-axis.Angles in the first quadrant are their own reference angles.Quadrantal angles (0°, 90°, 180°, 270°) do not have a reference angle. Ref Angle=60-0=60 Ref Angle=180-120=60 Ref Angle=240-180=60 Ref Angle=360-315=45.Example 1: a) Sketch the angle b) Shade the reference angle c) Find the measure of the reference anglea. 127°b. 353° c. 215°d. -115° Example 2: Given the reference angle and the specified quadrant, determine the value of the original angle, θ.a. reference angle =20°, Quadrant IIb. reference angle =45°, Quadrant IVc. reference angle =7°, Quadrant IIId. reference angle =62°, Quadrant IWe can use the special right triangles or chart from yesterday and the fact that each point is (cosθ, sinθ) to fill in the coordinate points for 30, 45, 60.If we know the coordinates in the first quadrant, we can determine all the other points.What do you notice about the points located at angles 30, 150°, 210°, 330°? Why do you think this is happening?All have the same reference angle! 30°=32,12Directions: Find the exact value of each trigonometric function. Express answers in simplest radical form, where appropriate.1. cos?(210°)2*. cos?(585°)Q 3585-360=225 =Q 3210-180=30 225-180=45cos210=-cos30=-32 cos585=-cos45=-223. csc?(-135°)4*. cot4π3-857252019301Finding the Exact Value at Any AngleSketch the angleDetermine the reference angleCalculate the trigonometric function value of the reference angleDetermine if the value of the function is positive or negative (ASTC)A useful way to remember this last step is “All Students Take Calculus”0Finding the Exact Value at Any AngleSketch the angleDetermine the reference angleCalculate the trigonometric function value of the reference angleDetermine if the value of the function is positive or negative (ASTC)A useful way to remember this last step is “All Students Take Calculus”Q 34?1803=240=Q 3180-135=45 240-180=60csc-135=1sin-135=1-sin45 cot4π3=1tan60=133=33=1-22=-22?22=-222=-2 1. Find the exact value of sec?(-240°) in simplest radical form.sec-240°(Q II)=-sec60°=-2secθ=1cosθ so -cos60°=-12 therefore-sec60°=-22. Find the exact value of tan5π6 in simplest radical form.tan5π6*180π=tan150°(QII)=-tan30°=-33Homework 13.6: Reference Angles and Exact Values1. Find the measure of the reference angle for each of the given angles:a. 100°b. 290°c. -130°2. If an angle is in the 3rd Quadrant and has a reference angle of 35°, what is the angle?3. Find the exact value of each of the following:a. tan315 b. sec?(-225)c. sin5π3d. sin510Lesson 13.7: Proving Pythagorean IdentitiesLearning Goals:What is a Pythagorean Identity?How can we use the Pythagorean identity to find the values of the sine, cosine, and tangent of an angle?Identities:In mathematics, an IDENTITY is an equation that is always true. Identities are very useful and can help us make “replacements” that may be useful when simplifying trigonometric expressions.229552571374000Recall the right triangle we made from the unit circle. Using this, we can prove the Pythagorean Identity:unit circle=radius of 1 x, y=(cosθ, sinθ) a2+b2=c2 x2+y2=12 cos2θ+sin2θ=12 So, the Pythagorean Identity is:cos2θ+sin2θ=1 What happens if you take the Pythagorean Identity and divide each term by cos2θ?What happens if you take the Pythagorean Identity and divide each term by sin2θ?cos2θ+sin2θ=1 cos2θcos2θ+sin2θcos2θ=1cos2θ 1+tan2θ=sec2θ cos2θ+sin2θ=1 cos2θsin2θ+sin2θsin2θ=1sin2θ cot2θ+1=csc2θ Use the Pythagorean identity to answer the following questions.1. An angle, α, has a terminal ray that falls in the second quadrant. If it is known that sinα=35, use cos2α+sin2α=1 determine the value of cosα.526732574993500Use: ASTC to determine the quadrant and if it is + or –cos2α+352=1 cos2α+925=1 cos2α=1625 cos2α=1625 cosα=±45, but must be-45 because it is in Q II 2. If cosθ=13 and tanθ<0, use the identity cos2θ+sin2θ=1 to find the vlaue of sinθ in simplest radical form.132+sin2θ=119+sin2θ=1 sin2θ=89 sinθ=±89 sinθ=±223 but only -so-223 5200650185420003. Suppose 0<θ<π2 and sinθ=13. What is the value of cosθ?Use: ASTC to determine the quadrant and if it is + or –0<θ<π2?180π so 0<θ<90 cos2θ+sin2θ=1 cos2θ+132=1 cos2θ+13=1 cos2θ=23 cos2θ = 23 cosθ=±23?33, but must be+63 because it is in Q I 4. If cosθ=-15, what are possible values of sinθ?5029200250825005. If sinθ=-23 and tanθ>0, find cosθ. Means sinθ=- and tanθ=+cos2θ+sin2θ=1 cos2θ+-232=1 cos2θ+49=1 cos2θ=59 cos2θ = 59 cosθ=±53, but must be-53 because it is in Q III 6. If the terminal ray of β lies in the fourth quadrant and sinβ=-33 determine cosβ and tanβ in simplest form.7. Use the Pythagorean identity sin2θ+cos2θ=1, where θ is any real number, to find the following:tanθ and sinθ, given cosθ=-513, for π<θ<3π2. or 180<θ<270 Q IIIcos2θ+sin2θ=1 -5132+sin2θ=1 25169+sin2θ=1 sin2θ=144169 sin2θ = 144169 sinθ=±1213, but must be-1213 because it is in Q III tanθ=sinθcosθ=-1213-513=125 8. If the terminal side of angle θ, in standard position, passes through point (5, -4), what are the numerical values of the six trig functions?Draw a right triangle using the point! It is in Q IV42+52=c2 sinθ=441=-44141cscθ=-41416+25=c2 cosθ=541=54141secθ=41541=c tanθ=-45cotθ=-549. If the terminal side of angle θ, in standard position, passes through point (-2, -5), what are the numerical values of the six trig functions?Homework 13.7: Proving Pythagorean Identities1. Use the Pythagorean identity sin2θ+cos2θ=1, where θ is any real number, to find the following:a. cosθ, given sinθ=513, for π2<θ<πb. tanx, given cosx=-12 for π<x<3π2c. cosθ and tanθ, given sinθ=13, for 0<θ<π2d. sinx and tanx, given cosx=-56, for π<x<3π22. Find the degree measures of two angles, one positive and one negative, that are coterminal with 65°.Lesson 13.8: Simplifying Trig ExpressionsLearning Goals:How do we simplify trig expressions?How do we prove trig identities?Do Now: In order to prepare for today’s lesson, write down the identities we have learned this unit.Pythagorean IdentitiesReciprocal IdentitiesQuotient Identitiescos2θ+sin2θ=1cos2θ=1-sin2θsin2θ=1-cos2θ1+tan2θ=sec2θtan2θ=sec2θ-1cot2θ+1=csc2θcot2θ=csc2θ-1cscθ=1sinθsecθ=1cosθcotθ=1tanθtanθ=sinθcosθcotθ=cosθsinθUsing the identities above, write an equivalent trig function for each of the following expressions below:a) 1-sin2θ=cos2θb) 1secθ=cosθc) sinθcosθ=tanθ-47625162560Simplifying Trig Expressions TIPSIf you have a 1-trig function2, try a Pythagorean identityTry rewriting all trig functions in terms of sinθ and cosθTry factoring out a GCF if you canDon’t forget about rules used with numbers:Dividing fractions → keep change flipSquaring a binomial → double distributiveMultiply by the conjugate0Simplifying Trig Expressions TIPSIf you have a 1-trig function2, try a Pythagorean identityTry rewriting all trig functions in terms of sinθ and cosθTry factoring out a GCF if you canDon’t forget about rules used with numbers:Dividing fractions → keep change flipSquaring a binomial → double distributiveMultiply by the conjugateExample 1: Simplify each expression.a) 1-cos2xcscxb) (cosθ)(secθ)sin2x1sinx=sinx?sinxsinx=sinx cosθ1cosθ=cosθcosθ=1c) tan2x-tan2xsin2xd) tan2θsin2θtan2θ=sin2θcos2θGCF =tan2xsin2θcos2θsin2θ=sin2θcos2θ?1sin2θtan2x(1-sin2x) 1cos2θtan2x(cos2x) sec2θsin2xcos2xcos2x1 sin2x e) cos2θ1-sinθ1-sin2θ1-sinθ (1-sinθ)(1+sinθ)1-sinθ 1+sinθ -47625152400Proving Trig IdentitiesShow that the left side of the equation equals the right side of the equationChange the more complicated side to look like the simpler side – do not change both sides!0Proving Trig IdentitiesShow that the left side of the equation equals the right side of the equationChange the more complicated side to look like the simpler side – do not change both sides!Example 2: Prove the following trigonometric identities, beginning with the side of the equation that seems to be more complicated. Make sure that the complete identity statement is included at the end of the proof.a) tanx=sec?(x)csc?(x) b) cotx+tanx=secxcsc?(x) =1cosx1sinx cosxsinx+sinxcosx= =1cosx?sinx1 cosx*cosxsinx*cosx+sinx*sinxcosx*sinx= =sinxcosx cos2x+sin2sinx*cosx tanx=tanx √ 1sinx*cosxsecxcscx=secxcsc?(x)√Example 3: Write each expression as a single trig expression.a) sin2θcosθtanθcscθb) cscθsecθ(sin2θ)(cosθ)sinθcosθ1sinθ 1sinθ1cosθsin2θ cosθsinθcotθ or 1tanθExample 4: Show that sec2x-1sec2x is equivalent to sin2xsec2x-1sec2x=sin2x 1cos2x-cos2xcos2x1cos2x common denominator of cos2x1-cos2x sin2x=sin2x√ Homework 13.8: Simplifying Trig ExpressionsDirections:1. tan2θsin2θ2. cotθcosθ3. sin?(x)cot?(x)4. sin2θ1-cosθ Verify the following identities:5. sec2θ-1sec2θ=sin2θ6. cosxcscxtanx=17. Error Analysis: Describe and correct the error in simplifying the expression.1-sin2θ=1-1+cos2θ =1-1-cos2θ =-cos2θ ................
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