MATH 127 SAMPLE FINAL EXAM I II III TOTAL

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1 MATH 17 SAMPLE FINAL EXAM Name: Section: Do not write on this page below this line Part I II III TOTAL Score

2 Part I. Multiple choice answer exercises with exactly one correct answer. Each correct answer is worth point. 1. Which of the following expressions is always equal with sin 5 θ + sin θ cos θ? a) sin 8 θ cos θ b) sin θ c)sin θ d)cot θ. Which of the following expressions is always equal with cos x 1? a) sin x b) cos x c) (cos x 1)(cos x + 1) d) (cos x 1). Which of the following expressions is always equal with a) cos x sin x b) cos x + sin x c) cos x sin x cos x sin x? 1 cos x + sin x d) cos x 4. Let x kπ, k integer. Then E = sin x cos x tan x csc x sec x is equal with (chose one): a) 1 b) c) sin x cos x d)1 + sec x 5. Which of the following expressions is always equal with a) 90 b) 90 A A sin(90 A) sin A c) tan A d) cot A 6. Which of the following trigonometric functions is undefined for an angle of 60 : a) sec b) cos c) cot d) tan? 7. Which of the following expressions is equal with sin 40 : a) 40 b) cos 50 c) sin 0 d) csc 40 ( 8. Which of the following expressions is equal with sin π ) 5? a) 1 5 b) cos π 5 c) sin π 5 d) csc 5 π 9. Which of the following expressions is equal with cos(4π t)? a) cos 4π cos t b) cos t c) cos t d) 4π cos t 10. A period of the function y = 17 sin(πx) is a) π b) 1 c) d) 17

3 11. The amplitude of the function y = 1 cos(πx) is π a) 1 π b) π c) π d) 1 π 1. Consider the function y = π tan(x). The value π is for this function: a) the phase shift b) the amplitude c) the y-intercept d) the period 1. Consider the equation sin x = 1 + cos x. Which of the following is a solution of this equation? a) x = π b) x = π c) x = 0 d) x = 1 ( 14. Which of the following real numbers is equal with arcsin 1 )? a) sin b) π 6 c) csc() d) 11π Which of the following points is on the parametric curve x = sin t, y = cos t? ( a) ( 1, 0) b) 1, 1 ) c) (1, 1) d) (, 0) 16. An angle of 150 converted in radians is a) 6π 5 b) π 5 c) 5π 6 d) 150π 17. An angle of π 5 radians converted to degrees is a) 6 b) 7 c) 18 d) Which of the following products is always equal with the sum sin(x) + sin(4x)? a) sin x cos 4x b) sin x cos x c) cos x sin x d) sin 6x cos x 19. Which of the following expressions is always equal with the product sin(x) sin(x)? a) cos 5x cos x b) sin x + sin 5x c) cos x + cos 5x d) cos x cos 5x 0. The curve of equation r = sin θ in polar coordinates, has in rectangular coordinates the form a) x + y = b) x + y = y c) ± x + y = arctan y x d) x + y y = 0

4 Part II. Each of the following exercises is worth points. Please write the answer in the place provided. For partial credit show your work. 1. Find the area of a triangle ABC in which the angle A is 0 and AB = in, AC = in.. Evaluate the trigonometric functions tan and csc of an angle of 600. tan( 600 ) = csc( 600 ) =. Evaluate the trigonometric functions sec and cot of an angle of 15π 4. sec 15π 4 = cot 15π 4 =

5 4. From a point level with the base of a tree, the angle of elevation to the top of the tree is 45. Knowing that the tree is 100 ft tall, determine the distance to the base of the tree. 5. In a circle of radius 7 find the length of the arc determined by a central angle of π 5 radians. 6. In the triangle ABC we know the sides AB = and BC =, and the angle B = 60. Determine the side AC.

6 7. If tan x = and tan y =, find tan(x + y). 8. Find a positive real number t such that cos t = 1 and a negative real number s such that cos s = 1. t = s =

7 Part III. Each of the following exercises is worth 4 points. Please write the answer in the place provided. Show your work for partial credit 1. Given cos B = a 5 and 90 < B < 180, determine sin B and cot B as functions of a. sin B = cot B =. Let ABC be a right triangle with the angle B = 90. If BC = 6 and AC = 10, determine the following quantities: AB, sin A, cos A and tan A. AB = sin A = cos A = tan A =

8 . Find three distinct x-intercepts of the function y = sin x. 4. If sin x = 1, where π < x < π, and cos y =, where π < y < π, determine sin(x y).

9 5. Find all real-number solutions of the equation tan x = If sin x = 1 and 0 < x < π, find cos x and tan x.

10 7. Let u = 1, and v =, 1. Determine the length of the vector u v. 8. Evaluate ( 1 + i ) Prove that sin x = tan x 1 + tan x is an identity.

11 MATH 17 FORMULAS θ in degrees θ in radians sin θ cos θ tan θ π π π 1 90 π 1 0 undefined sin θ + cos θ = 1 sec x = 1 + tan x tan θ = sin θ sec θ = 1 csc θ = 1 cos θ cos θ sin θ ( π ) ( π ) sin θ = cos θ cos θ = sin θ cot θ = cos θ sin θ sin( θ) = sin θ cos( θ) = cos θ tan( θ) = tan θ sin(θ + π) = sin θ cos(θ + π) = cos θ tan(θ + π) = tan θ sin(s + t) = sin s cos t + cos s sin t cos(s + t) = cos s cos t sin s sin t tan s + tan t tan(s + t) = 1 tan s tan t sin θ = sin θ cos θ cos θ = cos θ sin θ = 1 sin θ = cos θ 1 tan θ = tan θ 1 tan θ sin θ = 1 cos θ cos θ = 1 + cos θ sin a sin b = 1 [cos(a b) cos(a + b)] cos a cos b = 1 [cos(a b) + cos(a + b)] sin a cos b = 1 [sin(a + b) + sin(a b)] sin(s t) = sin s cos t cos s sin t cos(s t) = cos s cos t + sin s sin t tan s tan t tan(s t) = 1 + tan s tan t tan θ = sin θ 1 + cos θ sin a + sin b = sin a + b cos a + cos b = cos a + b cos a b cos a b sin a sin b = sin a b cos a cos b = sin a + b cos a + b sin a b