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Review for Exam II
MATH 1113 (Ritter)
Sections Covered: 6.2, 6.3, 6.4, 6.5, 7.1, 7.2, 7.3
(1) Use the known values of the trigonometric functions for the angles 30◦ , 45◦ and 60◦ to find
the exact value of:
(a) cos
π
3
tan
π
4
(b) sin(30◦ ) + 1
(c) cot(30◦ ) csc(60◦ )
(2) Use the given information about the angle θ in standard position to determine which quadrant
contains its terminal side.
(a) sin θ < 0 and tan θ < 0
(b) sec θ > 0 and cot θ > 0
(c) csc θ > 0 and cos θ < 0
(3) Use the known trigonometric values and reference angles as needed to evaluate each expression exactly (without using a calculator).
(a) cos
7π
6
(b) csc(300◦ )
(c) tan
(d) sin
4π
3
3π
4
(e) sec(7π)
(4) Express each trigonometric expression in terms of an equivalent cofunction value.
(a) sin (72◦ )
π
5
(b) sec
(c) tan(130◦ )
(5) One trigonometric function value of the acute angle θ is given. Find all of the remaining
trigonometric functions of θ.
(a) sin θ =
1
2
(b) sec θ = 3
(6) Let f (θ) = tan θ and g(θ) = csc θ. Determine the exact value of
(a) f
π
4
(b) g
π
6
(c) f
θ
2
(d)
f (θ)
2
if θ = 60◦
if θ = 60◦
(e) g(45◦ )
(7) Determine the period and amplitude of the function. Identify the phase shift if any (and
direction left/right) and the vertical shift if any (and direction up/down). Plot one full period of
the graph of the function. Label the x-axis at key points (intercepts, maxima, and minima) and
the y-axis at any vertical shift and maximum and minimum values.
πx −1
2
π
(b) y = 3−2 cos 2x −
2
(a) y = 2 sin
(8) Plot at least one full period of the graph of y = 2 csc
πx
2
− 1.
(9) Determine the period of the function. Identify the phase shift if there is one, and determine
if it is to the left or the right. Identify the vertical shift if there is one, and determine if it is up
or down. Identify one pair of adjacent vertical asymptotes of the function, and plot one period.
(a) y = tan(2x)+2
π
(b) y = 2 cot x −
4
(10) State the domain and range of each of the following functions.
(a) y = sin−1 x
(b) y = 2 tan−1 x
(c) y = cos−1 (2x)
(11) Evaluate each expression exactly (without using a calculator).
√ !
3
−1
−1
−1
√ ,
(a) sin
,
(b) tan
(c)
2
3
(d)
−1
cos sin
1
,
3
(e)
−1
tan
cos−1 0,
7π
tan
6
(12) Express each of the following as a purely algebraic expression.
(a)
sin(sec−1 x),
(b)
x cos tan−1
,
2
(c)
cos sin−1 x − tan−1 x
(13) Use an appropriate sum, difference, double, or half angle formula to evaluate each expression exactly (without a calculator).
(a)
(b)
(c)
(d)
sin(217◦ ) cos(37◦ )−cos(217◦ ) sin(37◦ )
tan(40◦ ) + tan(20◦ )
1 − tan(40◦ ) tan(20◦ )
5π
cos
8
sin(15◦ )
(14) Use the given information to evaluate the quantities exactly.
2
sin α = ,
3
0<α<
π
,
2
and
(a) cos(α + β)
(b) sin(α − β)
(c) tan(2α)
(d) csc
α
2
(15) Verify the given identities.
(a)
sec4 θ−tan4 θ = 2 tan2 θ+1
(b)
cot x+tan x =
sec2 x
tan x
(See next page for information about formulas)
tan β = 5 π < β <
3π
2
I will provide the following formulas. Any formulas not listed here will not be provided.
Sum of Angles Formulas
sin(u + v) = sin u cos v + sin v cos u
cos(u + v) = cos u cos v − sin u sin v
tan(u + v) =
tan u + tan v
1 − tan u tan v
Half Angle/Power Reducing Formulas
sin x =
1−cos(2x)
,
2
cos2 x =
tan2 x =
2
±
q
1−cos x
2
±
q
1+cos x
2
sin
x
2
=
1+cos(2x)
,
2
cos
x
2
=
1−cos(2x)
,
1+cos(2x)
tan
x
2
q
= ± 1−cos(2x)
1+cos(2x)
Note: No calculator use will be allowed. I encourage you to work this review without relying on
one. Remember that there are also suggested exercises from the text listed on the class website.