Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan. Review Problems for Test #3

Transcription

1 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 π 3 ). Exercise 3 Find the period and the phase shift of y = 3 cot ( x + 3π). Exercise Given sin α = 12, α in quadrant III, and cos β =, β is qaudrant II. Find 5 13 sin (α β). Exercise 5 Find an equation of a simple harmoninc motion with frequency f = 1 cycles 2π per second and amplitude 5 inches. Assume maximum displacement occurs at t = 0. Exercise 6 Establish that 1+sin x cos x = 0. cos x 1 sin x Exercise 7 Establish the identity: sin ( x + π 2 ) = cos x. 1

2 Exercise 8 State the amplitude, period, and phase shift of y = 5 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: Exercise 11 Find the exact value of: sin 7x cos 3x cos 7x sin 3x. tan ( 7π 12 ) tan ( π ) 1 + tan ( 7π 12 ) tan ( π ). Exercise 12 Find the exact value of sin 2θ, cos 2θ, and tan 2θ given that cos θ = 0 and θ 1 in Quadrant IV. Exercise 13 Write y = 1 sin x 3 cos x in the form k sin (x + α) k and α are to be 2 2 determined. Exercise 1 Find the exact value of sin α, cos α, and tan α Quadrant I. given that sec α = and α in Exercise 15 Write the following expression as a product of two trigonometric functions: cos θ cos θ. 2 Exercise 16 Use the trigonometric identities to write the expression 1 1 sin t sin t in terms of a single trigonometric function. 2

3 Exercise 17 Graph one full period of the function f(x) = 3 cos (5x). Exercise 18 Find the value of sin θ given that sec θ = 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 2 x). 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) = 2 sin ( x ) 2π 3 3. Exercise 23 Find an equation of the graph Exercise 2 Find the equation of the cosine function with amplitude 3, period 3π, and phase shift - π. 3

4 Exercise 25 Show that tan x sec x = tan 2 x + sec 2 x is not an identity. Exercise 26 Find the amplitude, phase shift, and period of y = sin x 2 cos x 2. Exercise 27 Find the exact value of sin x 2 quadrant. given that cot x = 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 θ 2 + cos θ

5 Exercise 3 Establish the identity. (a) sin 5x cos 3x = sin x cos x + sin x cos x. (b) 2 cos 5x cos 7x = cos 2 6x sin 2 6x + 2 cos 2 x 1. Exercise 35 Write each expression as the sum or difference of two functions. (a) 2 sin x 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 (b) sin π 12 7π cos (c) sin 11π 7π sin Exercise 37 Use a half-angle formula to find the exact value of sin Exercise 38 Find tan x 2 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 0 Write sin 7x + sin 3x as a product of trigonometric functions. 5