Sarah Brewer - Alabama School of Math and Science



Trigonometry Final Exam Study GuideGeneral Study Tips:?rework all tests and take-home quizzes, example problems from class, and tests in online test archive?memorize formulas and names of formulas, and the cases to which you apply themTopics to Know:Algebra?manipulate fractions, including complex fractions?rationalize denominators?factor?apply the square root theorem, zero product property, and quadratic formulaBasic Trig Topics (Ch. 5):? evaluate without a calculator all six trig functions of any angle in radians or degrees having a 0, 30, 45, 60, or 90 reference angle?evaluate without a calculator all six trig functions of any angle, given one of the six trig function values and the quadrant?apply concepts of cofunction and like reference angles to evaluate trig functions given angles that are complements or have the same reference angle but are in different quadrants?convert between radians and degrees?angles of elevation and depression?linear speed, angular speed, and arc lengthGraphing:?graph all 6 basic trig functions? find amplitude and period?rewrite all equations by factoring to find horizontal shift (-c/b)?apply horizontal and vertical shifts?graph sum and difference of basic trig and linear functionsIdentities (Ch. 6):?evaluate trig functions of a given angle using sum, difference, half, and double angle identities?simplify expressions using identities?evaluate double, half, sum, and difference identities of an angle, given trig functions of one or more original angles and the quadrant(s)?evaluate inverse trig functions and compositions of inverse functions?solve trig equations for both all real values of the variable and for all values of the variable in a given interval?prove trigonometric identitiesSolving Triangles (Ch. 7)?identify type of triangle (SSS,SAS,SSA,ASA,AAS)?solve any triangle given at least one side and two other measures?solve word problems involving the Law of Sines and Law of Cosines?find the area of any triangleVectors (Ch. 7)?perform arithmetic operations with vectors (scalar multiplication and vector addition/subtraction)?find the magnitude and direction angle of any vector?write vectors in component form and in terms of i and j?find a unit vector in the direction of any vector?find the dot product of two given vectors?find the angle between two vectors?solve word problems involving a mass on an incline?solve word problems involving heading of boats and airplanesTrigonometric Form of Complex Numbers (Ch. 7)?convert between standard form and trigonometric form?determine the modulus and argument of a given complex number?multiply and divide complex numbers in trigonometric formTrig Functions of an Acute Anglesinθ=side opposite θhypotenusecscθ=hypotenuseside opposite θcosθ=side adjacent to θhypotenuse secθ=hypotenuseside adjacent to θtanθ=side opposite θside adjacent to θ cotθ=side adjacent to θside opposite θConverting Between Degree & Radian MeasureTo convert from degree to radian measure, multiply by π180°To convert from radian to degree measure, multiply by 180°πArc Length and Angular SpeedVariabless=distance traveled or arc length inches, kilometers, etct=time (seconds, minutes, hours, days, etc)θ=amount of rotation or included angle (degrees, radians, rotations, revolutions, etc)r=radius or distance from the center of rotation (centimeters, inches, etc)v=linear speed=distancetimeω=angular speed=amount of rotationtimeFormulass=rθ , v=st , ω=θt , v=rωDimensional analysis conversion factors5280 ft1 mi , 12 in1 ft , 2π1 rev , π180° , 60 min1 hr , 60 sec1 min , and their reciprocalsExample problem: A car travels at 60 miles per hour. Its wheels have a 24-inch diameter. What is the angular speed of a point on the rim of a wheel in revolutions per minute?Solution: v=60 mi1 hr , =12 in , ω= ? Equation relating these variables: v=rωω=vr=v?1r=60 mi1 hr?112 in?1 hr60 min?5280 ft1 mi?12 in1 ft?1 rev2π=2640πrevminReciprocal Identitiescscx= 1sinxsinx= 1cscx secx= 1cosx cosx= 1secx cotx= 1tanx tanx= 1cotxRatio Identitiestanx= sinx cosx cotx= cosxsinx Pythagorean Identitiessin2x+cos2x=1 , 1+cot2x=csc2x , tan2x+1=sec2xOdd-Even Identitiescos-x=cosx , sin-x=-sinx , tan-x=-tanxsec-x=secx , csc-x=-cscx , cot-x=-cotxSum and Difference Identitiessina+b=sinacosb+cosasinb sina-b=sinacosb-cosasinbcosa+b=cosacosb-sinasinb cosa-b=cosacosb+sinasinbtana+b=tana+tanb1-tanatanb tana-b=tana-tanb1+tanatanb Cofunction Identitiessinπ2-x=cosx , cosπ2-x=sinxtanπ2-x=cotx , cotπ2-x=tanxcscπ2-x=secx , secπ2-x=cscxDouble-Angle Identitiessin2x=2sinxcosxcos2x= cos2x-sin2xcos2x=2cos2x-1cos2x=1-2sin2x tan2x=2tanx1-tan2x Half-Angle Identitiessinx2=±1-cosx2 cosx2=±1+cosx2 tanx2=±1-cosx1+cosx tanx2=sinx1+cosx tanx2=1-cosxsinx Solving TrianglesLaw of SinessinAa=sinBb=sinCc or asinA=bsinB=csinCLaw of Cosinesc2=a2+b2-2abcosCb2=a2+c2-2accosBa2=b2+c2-2bccosAArea of a TriangleK=12bcsinA=12acsinB=12absinCVectorsThe component form of AC with A=(x1, y1) and C=(x2, y2) is AC=x2-x1, y2-y1The magnitude of a vector v with component form a, b is v=a2+b2The reference angle α for the direction angle θ of the vector a, b is given by α=tan-1ba. Figure outwhich quadrant this angle should be in and measure the angle counterclockwise from the positive x-axis. The horizontal component of the vector a, b is a=vcosθThe vertical component of the vector a, b is b=vsinθFor a real number k and a vector v=v1, v2, the scalar product of k and v is kv=kv1, v2=kv1, kv2. The vector kv is a scalar multiple of the vector v.Vector Addition/Subtraction: If u=u1, u2 and v=v1, v2, then u±v=u1±v1, u2±v2 .If v is a vector and v≠0, then vv is a unit vector (vector with magnitude 1) in the direction of v.The dot product of two vectors u=u1, u2 and v=v1, v2 is u?v=u1v1+u2v2 .If θ is the angle between two nonzero vectors u and v, then cosθ=u?vuv .Trigonometric Form of Complex NumbersA complex number z=a+bi, where i=-1 can be written in trigonometric form as z=rcosθ+isinθ or z=rcisθ, where r=a2+b2 is the modulus of z and direction angle θ is referred to as the argument.z1z2=r1r2cosθ1+θ2+isinθ1+θ2=r1r2cisθz1z2=r1r2cosθ1-θ2+isinθ1-θ2=r1r2 cisθzn=rncosnθ+isinnθ=rn cisnθ ................
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