Trigonometry Semester Exam Review Name: 1. Give a real life example of a situation that can be modeled with a periodic function.. As a child goes up and down on a seesaw, his or her distance form the ground depends on time. Sketch a reasonable graph of this function. 3. Name the angle whose measure is between 0 and 360 that is coterminal with 4897. 4. Find the reference angle for each of the following: a) 133 b) 54 c) 317 d) 71 5. Find the six trigonometric functions for an angle whose terminal side passes through the point (-5, ). 6. Find the six trigonometric functions of θ if θ terminates in Quadrant III and sin θ=. 3 7. Suppose you have been assigned the job of measuring the height of the local water tower. Climbing makes you dizzy, so you decide to do the whole job at ground level. From a point 47.3 meters from the base of the tower, you find that you must look up at an angle of 53 to see the top of the tower. How high is the tower?
8. Find θ if θ = sin -1 0.4791. 9. Find sin 7.4 ; find cot 3. 10. From a window 0 meters high, the angle of depression to the top of a nearby streetlight is50 o. The angle of depression to the base of the streetlight is58 o. How tall is the streetlight? 11. Given: y = C + A sin B (θ D) Explain the effects of A, B, C, and D. 1. Graph y = -5 + 7 cos 30 (θ + 4 ) Trig: VS: HS: Reflect? Start: Amp: Period: CP: 13 What is the highest point on the graph of y = 15+ cos4( θ 9)?
14. Sketch a quick graph of each of the following in radians: a) y = sin x d) y = cot x b) y = cos x e) y = csc x c) y = tan x f) y = sec x 15. Write the equation of the graph in radians using sin: 5 y -17-7 3 13 3 33 43-10 x -45 16. Convert to radian measure: a) 45 b) 10 c) 150 d) 5 e) -90 17. Convert to degree measure: 4 a) b) c) 3 6 7 4 d) 5 6 e) 5 18. Find the exact value of: 3 a) sin b) tan c) cos 3 4 d) cos 6 19. If x = cos -1 0.7, then x =? in radian measure.
0. Find the exact principal value of each in degrees: a) θ = Sin -1 1 b) θ = Cos -1 3 c) θ = Sin -1 1. Find the exact principal value of each in radians: a) Arcsin 1 1 b) x sec ( ) 1 = c) x = tan ( 0). Find the exact value of the expression using radians and radicals if necessary. a) sin cos 6 1 5 b) sin cos 4 1 3 c) cot sec 3 1 3. Given: f(x) = + 3 cos 9 (x 6) a) Find f(8) correct to three decimal places b) Find the first three positive values of x for which f(x) = 1.3. 4. Given f(x) = 5 + 4 sin (x + 10) 1 a) Find f(1) b) Find the first three positive values of x for which f(x) =.
Use the following information for problems 5-9 You are sitting on the deck of a river steamboat. As the paddlewheel turns, a point on the paddle blade is moved in a way such that its distance, d, from the water s surface is a sinusoidal function of time, t. When your stopwatch reads 7 seconds, the point is at it s highest, 0 feet above the water s surface. The wheel s diameter is 4 feet, and it completes a revolution every 16 seconds. 5. Sketch a graph of the sinusoid. 6. Write the equation of the sinusoid. 7. How far above the surface of the water is the point when your stopwatch reads seconds? 8. What is the FIRST positive value of time at which the point is at the water s surface? 9. At that first positive value, Is the point going into the water or coming out of the water? 30. Prove: (1 + cos x)(1 cos x) = sin x
31. Prove: cot x + tan x = csc x sec x 3. Prove csc θ cos θ + sin θ = csc θ 33. Use the cofunction properties to find an equivalent expression. a) cot 46 o b) sec 3 c) cos 4 1 35. Angles A and B are in standard position. Letsin A =, cos A < 0 and tan B =, cos B > 0. Draw 3 3 angles A and B in the appropriate quadrants and find the following: a). sin ( A+ B) b). cos( A+ B) c). tan ( A+ B)
36. Find the exact value of sin 75 o 37. Find the value for cos15 o Solve each of the following equations for the general solution 38. o o 1 sinθ cos 31 + cosθ sin 31 = 39. tan x tan x 1 = 1+ tan x tan x 3 40. tanθ + 1 = 0 41. cos x+ cos x= 0
4. sin x 5sin x 6 0 sin θ + 46 = o + + = 43. ( ) 44. cos 3θ cos1 o o 1 sin 3θ sin1 = 45. sin x+ 5sin x+ 6 = 0 46. 4cos x = 1