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MATH 116 EXAM 4 Key NAME (Print your name): You need to show all your work to get a full credit. to degree. Problem 1(5 point each) a) Convert 5 4 solution) 5 180 900 225 degree. 4 4 b) Convert 4 radian to degree. solution) 720 4 180 degree. Problem 2(7 point each) a) Let 3 . Find the reference angle and sin , cos , tan . solution) We see that the angle is in the Quad 1. The reference angle is 3 or 60 degree. So, we have a triangle with hypotenuse 2, opposite 3 and adjacent 1. Hence, sin opp 1, 2 hypo cos adj 1, 2 hypo tan opp adj 3 1 3, b) Let 135 0 . Find the reference angle and csc , sec , cot . solution) Move 135 degree from the positive x-axis clockwise. So, we have a triangle on the Quad 3. And we see that the reference angle is 45 degree. The hypotenuse is 2 , opposite is 1 and the adjacent is 1. Thus, csc hypo 2 2. 1 opp sec hypo 2 2. 1 adj adj 1 1. 1 opp cot 1 3 Problem 3(7 point each) Let sin x and cos x 0. a) Find cos x and tan x by drawing method. solution) Since cos is positive and sin is negative, we know that the angle x is in the Quad 4. We also know that hypotenuse is 3 and the opposite is 1, since sin x 1 . Now, the adjacent is 3 31 2. But x is in the Quad 4. So, the adjacent is 2 . Thus, 2 3 cos x 6 . , 3 tan x 1 . 2 b) Find cos x and tan x by using identities. solution) sin 2 x cos 2 x 1. So, cos 2 x 1 sin 2 x. 2 cos x 1 1 3 cos x 2 1 1 2. 3 3 2 . 3 Since cos is positive, we get cos x 2 3 2 3 To find tan x, let’s use sin x . tan x cos x 6 3 . tan x 1 3 2 3 1 3 1 . 2 3 2 Problem 4 a) (6 points) Sketch y sin x over two periods. solution) Note: You should indicate x 0, 2 , , 3 , 2 on the x-axis. 2 y 1 0.5 0 -5 -2.5 0 2.5 5 x -0.5 -1 b) (6 points) Find sin and sin solution) 3 2 sin 0, . sin 3 1. 2 c) (6 points) Sketch the graph of csc x and find the period. solution) The period is 2. Vertical asymptoes are where sin x 0. So, x 0, , 2, . . . . Here, the vertical lines are the vertical asymptotes x , 0, . y 5 2.5 0 -5 -2.5 0 2.5 5 x -2.5 -5 d) (8 points) Let y 5 sin 13 x. Sketch the graph over two periods. What are the amplitude and the period? solution) The amplitude is 5 and the period is 2 2 3 6. 1 3 y 5 2.5 0 -15 -10 -5 0 5 10 15 x -2.5 -5 e) (8 points) Sketch y sinx . Also find an identity for sinx . (You may use one of the following identities: sinA B sin A cos B sin B cos A, cosA B cos A cos B sin A sin B.) solution) By translating the graph of sin x to the right by , we have the following graph. y 1 0.5 0 -5 -2.5 0 2.5 5 x -0.5 -1 You may find an identity sinx cos x by examining the graph. Or, by making use of the identity given, we have sinx sin x cos sin cos x sin x. f) (4 points) Is sinx sin x? Explain your answer. solution) No, since the graph is symmetric with respect to the origin (that means odd function, so sinx sin x). Problem 5 (6 points) Sketch the graph of y tan x and find the period. Also find the period of y tan 4x. solution) The period of tan x is and the period of tan 4x is . 4 . . . . Vertical lines are the vertical asymptotes x 2 , 2 , 3 2 y 50 25 0 -1.25 0 1.25 2.5 3.75 5 x -25 -50 Problem 6(6 point each) a) Show that sin 3 x sin 5 x cos x sin 3 x cos 3 x. solution) Here the useful identity is sin 2 x cos 2 x 1. sin 3 x sin 5 x cos x sin 3 x1 sin 2 x cos x sin 3 x cos 2 x cos x sin 3 x cos 3 x. b) Substitute x 3 sec and simplify x 2 9 solution) The useful identity in this problem is 1 tan 2 sec 2 . x2 9 9 sec 2 9 9sec 2 1 3 sec 2 1 2 tan 2 2 tan . 4 4 c) Simplify cos x sin x as much as possible (hint: factor the numerator). cos x sin x solution) Note that cos 4 x sin 4 x cos 2 x sin 2 xcos 2 x sin 2 x cos 2 x sin 2 xcos x sin xcos x sin x. Since cos 2 x sin 2 x 1, we have cos 4 x sin 4 x cos x sin xcos x sin x. Thus, cos 4 x sin 4 x cos x sin xcos x sin x cos x sin x. cos x sin x cos x sin x