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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