I. State the equation of the unit circle. MATH 111 FINAL EXAM REVIEW x y y = 1 x+ y = 1 x = 1 x + y = 1 II. III. If 1 tan x =, find sin x for x in Quadrant IV. 1 1 1 Give the exact value of each expression. 1 1. sin 1 0 Undefined. cos 1 1 0 Undefined 1. sin10. cos 660 1 1 1 1
. cos 6. cos 0 1 1 0 1 Undefined 7. tan 1 8. cot 1 1 1 1 0 Undefined 9. sec 6 10. csc0 Undefined 0 1 IV. Which of the following is a sketch of the graph of the given function on [0, )? 1. y = sin x
. y = cos x. y = tan x
. y = cot x. y = sec x
6. y = csc x
V. Simplify each expression. 1. cos( θ ) cosθ cosθ sinθ sinθ. sin( θ ) cosθ cosθ sinθ sinθ. tan( θ ) tanθ tanθ cotθ cotθ. sec( θ ) secθ secθ cscθ cscθ VI. Evaluate each expression. 1. arccos 7. 1 sin
. 1 csccot 1. tan ( 1) 7 6 VII. Which of the following is a sketch of one cycle of the graph of each function? 1. y = sin x. y ( x ) = cos + +
. y = tan x
. y = cot ( x+ ). y = sec x
6. y = csc( x ) 1
VIII. Use the sum or difference identities to evaluate each expression. 1. cos 7 6 6 + 6 6+. sin 8 6 6 6 e. None of these. tan19 + 1 6+ IX. Let α be in Quadrant I, β in Quadrant III, 7 cosα =, and tan β =. 1 1. ( α β) cos =? 6 0 0. ( α β) sin + =? 0. ( α β) tan + =? 9 119 0 17
6 X. Change each sum or difference to a product. 1. sin 68 + sin cos0 cos18 sin 0 sin18 cos0 sin18 sin 0 cos18. sin x sin x cos xsin x sin x cos8x sin x cos xsin x. cos1x+ cos x cos17x 17x 7x sin sin 17x 7x sin cos 17x 7x cos cos. cos 0 cos 0 sin10 sin10 cos 0 cos10 XI. Let θ be in Quadrant II with 1 secθ =. 1. sin θ =? 1 10 169 60 169 10 e. None of these 1
. cos θ =? 1. tan θ =? 10 119 10 1 119 169 1 119 169 10 119 XII. Evaluate each of the following expressions using the half-angle identities. 1. sin11. 1+ + 1. cos17. 1+ + +. tan 67. 1 + + 1 1 XIII. If the terminal side of θ passes through the point (-,), find sin θ. 1 1 1 1 XIV. Solve each equation for 0 x <. 1. cos x= 1 sin x 1 1
7 11 0,,, 0,,, 0,,, 6 6 6 6, 6 6. 1 sin xcos x=,,,,. cos sin 1 0 x+ x =,, 6 6,, 6 6,, 7 11,, 6 6. cot x 1 = 0 7 11,,,,,,, 6 6 6 6 7,,, 6 6 11,,, 6 6 7 11 1 17 19,,,,,,, 6 6 6 6 6 6 6 6 XV. Solve ABC for the missing part. 1. A= 90, a = 9, b= 1, B=?.6 6..9.1. a =, b= 8, c= 10, C =? 9.7 97.9. 7.9
. A= 0, b= 6, B= 0, c=? 1. 8.1 11..6 XVI. Give the radian measure of an angle that subtends an arc of length in a circle of radius8. 1 19 None of these. XVII. Convert to degrees. 1 7 1 XVIII. Convert 60 to radians. 9 1 1 9 XIX. Simplify each expression. 1. sinθsecθ 1 9 6800 cotθ 1 sin θ tanθ. cos θ tan θ + cos θ 1 cot θ cos θ tan θ sin θ e. None of these. cscθ + secθ sinθ + cosθ 1 sinθ + cosθ csc θ + sec θ cscθsecθ
. ( ) sin x+ cos x sin x 1 1+ sin x sin xcos x. sec sec x tan x+ tan x x tan x 1 sec x tan x 1 6. 1 sin θ cotθ cos θsinθ cotθ secθ cosθsinθ cotθ XX. Change the product to a sum. 1. 6sin1 sin + +. sin xcos x cosx+ cos x sin x+ sin x cosx cos x cos x+ cos x. cos 8 sin 0 sin 68 sin1 1 1 sin 68 sin1 1 1 sin 68 + sin1 cos 68 + cos1
. cos 7xcosx 1 1 cos1x+ cos x 1 1 sin1x+ sin x 1 1 cos1x cos x 1 1 sin1x sin x XXI. Let the point 1, be a point on the terminal side of an angle θ in standard position. Find the sine and cosine of θ. 1 cos θ = ;sinθ = 1 sin θ = ;cosθ = 1 cos θ = ;sinθ = cosθ = ;sin θ = 1 1 XXII. For each of the following, give the quadrant in which the terminal ray of θ lies. 1. tanθ < 0 and cosθ > 0 I II III IV. cscθ > 0 and cot θ<0 I II III IV XXIII. Give the reference angle for the indicated angle. 1. 11 19 9 1
91. 9 18 9 19 18.. 9. +.. +. XXIV.Find the quadrant in which the indicated angle lies. 1. 1. 78 I II III IV I II III IV.. I II III IV. 1 I II III IV
XXV. Which of the following angles are coterminal with the given angle? 1. 17 677 7 7. 1 1 7 1 1 1 1 1 1 7 XXVI.Give the amplitude of the function f ( x) = 7 cos x+ + 1-7. XXVII.Give the period of the function f ( x) = 8sin 9 x + 8. 9 9 XXVIII.Give the period of the function f ( x) = tan x+ + 6. XXIX.Given the following data set for ABC, how many triangles can be drawn?
1. a = 1, b= 0, A= 1 0. a = 8, b= 1, A= 1 0 XXX. If cosθ =, θ in Quadrant III, find the value of tanθ. 7 6 7 6 6 7 6 XXXI. The length of an arc of the unit circle is as given. Name the quadrant within which the terminal point would lie. 1. t = I II III IV.. 1 t = 1 I II III IV.. t = 0 9 I II III IV.
. t =.78 I II III IV. XXXII.Give the terminal point on the unit circle for an arc of the length below. 1. t = 7 6 1, 1, 1, 1, None of these.. t =,,, ( 0, 1) None of these.. t = 1, 1, 1, 1, XXXIV.Complete the following statements: 1. 1 sin θ = tan θ sinθ cosθ
cos θ. sec θ tan θ = secθ tanθ sinθ 1 cos θ. = cos 7x sin 7 x 1 sin1x cos1x 0. 1 cos 0 = cos sin sin100 cos100. 1+ cot 9 x = csc 9x cot 10x sec 9x cos 9x 6. cos ( θ ) + = cosθ sinθ sinθ cosθ 7. ( θ ) sin + = cosθ sinθ sinθ cosθ
ANSWERS: I. c II. d III. 1. d. a. c. c. a 6. a 7. b 8. d 9. d 10. a IV. 1. b. a. a. b. b 6. a V. 1. b. d. a. b VI. 1. d. c. d. b VII. 1. d. c. c. a. a 6. a VIII. 1. d. c. a IX. 1. b. c. d X. 1. d. d. d. a XI. 1.. d. a XII. 1. b. c. c XIII. b XIV. 1. a. d. c. a XV. 1. b. b. a XVI. b XVII. a XVIII. b XIX. 1. d. a. d. c. d 6. c XX. 1. a. b XXI. a XXII. d XXIII. 1. c. d. a XXIV.1. b. a. d. b XXV. 1. b. a XXVI. b XXVII. d XXVIII. b XXIX. 1. b. d