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Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan Review Problems for Test #3 Exercise 1 The following is one cycle of a trigonometric function. Find an equation of this graph. Exercise 2 Sketch one full period of y = 2 cos 2x − π3 . Exercise 3 Find the period and the phase shift of y = −3 cot x 4 + 3π . Exercise 4 12 Given sin α = − 54 , α in quadrant III, and cos β = − 13 , β is qaudrant II. Find sin (α − β). Exercise 5 1 Find an equation of a simple harmoninc motion with frequency f = 2π cycles per second and amplitude 5 inches. Assume maximum displacement occurs at t = 0. Exercise 6 Establish that 1+sin x cos x − cos x 1−sin x = 0. Exercise 7 Establish the identity: sin x + π2 = cos x. 1 Exercise 8 State the amplitude, period, and phase shift of y = 54 cos (3x − 2π). Exercise 9 A harmonic motion is given by the formula f (t) = 2 sin 2t. Find the period and the frequency. Exercise 10 Write the following expression in terms of a single trigonometric function: sin 7x cos 3x − cos 7x sin 3x. Exercise 11 Find the exact value of: tan ( 7π ) − tan ( π4 ) 12 . 1 + tan ( 7π ) tan ( π4 ) 12 Exercise 12 Find the exact value of sin 2θ, cos 2θ, and tan 2θ given that cos θ = in Quadrant IV. Exercise 13 Write y = 21 sin x − determined. 40 41 and θ √ 3 2 cos x in the form k sin (x + α) k and α are to be Exercise 14 Find the exact value of sin α2 , cos α2 , and tan α2 given that sec α = Quadrant I. 17 15 and α in Exercise 15 Write the following expression as a product of two trigonometric functions: cos 2θ − cos θ. Exercise 16 Use the trigonometric identities to write the expression 1 1 + 1 − sin t 1 + sin t in terms of a single trigonometric function. 2 Exercise 17 Graph one full period of the function f (x) = − 34 cos (5x). Exercise 18 Find the value of sin θ given that sec θ = √ 2 3 3 and 3π 2 < θ < 2π. Exercise 19 Sketch one full period of the function y = −2 sec πx. Exercise 20 Graph one full cycle of the function f (x) = 3 csc π x 2 . Exercise 21 Graph f (x) = 3 tan πx for −2 ≤ x ≤ 2. Exercise 22 Find the amplitude, period and the phase shift of the function f (x) = x 2π −2 sin 3 − 3 . Exercise 23 Find an equation of the graph Exercise 24 Find the equation of the cosine function with amplitude 3, period 3π, and phase shift - π4 . 3 Exercise 25 Show that tan4 x − sec4 x = tan2 x + sec2 x is not an identity. Exercise 26 Find the amplitude, phase shift, and period of y = sin x2 − cos x2 . Exercise 27 Find the exact value of sin x2 given that cot x = quadrant. 8 15 with x in the third Exercise 28 Find an equation of a simple harmoninc motion with frequency f = 0.5 cycles per second and amplitude 5 inches. Assume maximum displacement occurs at t = 0. Exercise 29 Write an equation for the simple harmonic motion whose amplitude is 3 centimeters and period is 1 second assuming zero displacement at t = 0. Exercise 30 Write the given equation in the form y = k sin (x + α), where α is in degrees. (a) y = − √ sin x − cos x (b) y = 3 sin x − cos x. Exercise 31 Write the given equation in the form y = k sin (x + α), where α is in radians. (a) y = 2 sin √ x + 2 cos√x (b) y = − 2 sin x + 2 cos x. Exercise 32 Write each expression as the product of two functions. (a) sin 5θ + sin 9θ (b) cos 3θ + cos 5θ Exercise 33 Write each expression as the product of two functions. (a) sin 7θ − sin 3θ (b) cos 2θ + cos θ 4 Exercise 34 Establish the identity. (a) sin 5x cos 3x = sin 4x cos 4x + sin x cos x. (b) 2 cos 5x cos 7x = cos2 6x − sin2 6x + 2 cos2 x − 1. Exercise 35 Write each expression as the sum or difference of two functions. (a) 2 sin 4x sin 2x (b) 2 sin 5x cos 3x (c) cos 6x sin 2x Exercise 36 Find the exact value of each expression. (a) sin 105◦ cos 15◦ π cos 7π (b) sin 12 12 (c) sin 11π sin 7π 12 12 Exercise 37 Use a half-angle formula to find the exact value of sin 22.5◦ . Exercise 38 Find tan x2 if sin x = 2 5 and x is in quadrant II. Exercise 39 Express sin 3x sin 5x as a sum of trigonometric functions. Exercise 40 Write sin 7x + sin 3x as a product of trigonometric functions. 5