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Please note that NOTA = none of the above
(1) Find z to be the nearest
10
1
of a degree
a) 41.
o
b) 48.
o
c) 36.
o
d) 53.
o
e) NOTA
(2) Find the length of side p. Round to two places.
a) 4.09 cm b) 4.39 cm c) 5.60 cm d) 3.49 cm e) NOTA
(3) Convert 600 degrees to radians.
a)
3
10
b)
9
c)
9
d)
9
e) NOTA
(4) Convert 4.3 radians to degrees. Round to the nearest degree.
a) 14º b) 774º c) 493º d) 126º e) NOTA
(5) Find the period of y = −4 cot 2 x
a) π b)
2
c) 2π d)
4
e) NOTA
4 cm
3 cm
z
p
6 cm
o
Math 111 Review Problems
Multiple Choice
(6) Find the phase shift of the function y = −4 sin(2π x + π)
a)
4
b)
2
1
c)
2
1
d)
4
e) 2π
(7) The solutions of cos
2
x – cos x – 2 = 0 are (where k denotes an arbitrary integer)
a) 2k π b)
2
- 2 k π c) π + 2 k d) −π + 2 k π e) k π
(8) The exact value of csc
3
is
a) 2 b)
2
3
c)
2
1
d) –2 e)
3
2
(9) An angle measured in standard position has the point (4,-5) on its terminal ray.
What is cos (θ)?)?
a)
41
4
b)
41
5
c)
41
5
d)
41
4
e)
5
4
(10) Simplify the expression
1
cos
1
2
a
using fundamental identities. The result is
a) a
2
cot b) a
2
sec c) 0 d)
2
tan e) NOTA
x
x
tan
sec 1
2
a) 1 b) tan x c) x
3
tan d) cot x e) x
2
cot
(12) Given that
2
3
cos and θ)? is acute, what is value of θ)??
a) 30º b) 45º c) 60º d) 15º e) –30º
a)
y 2 3 cos( 2 x 2 ) b)
y 1 2 cos( x )
c)
y 1 2 sin( x ) d)
y 2 3 sin( 2 x 2 )
e) NOTA
(19) The graph shows
y 3 sin Bx C
. B =
a) 1 b) π c)
2
d)
2
e)
2
1
(20) Find the equation of the graph
a) y x
2
tan
b)
y tan 4 x
c)
y tan 2 x d) y x
2
1
tan
e) NOTA
(21) Given the figure, find x
a) 12 b) 40 c) 20 d) 48 e) 36
(22) A central angle of 63º is in a circle of radius 18cm. How long is the arc cut by the angle?
Round to 2 places.
a) 19.79 cm b) 9.90 cm c) 39.58 cm d) 79.16 cm e) NOTA
(23) A pole casts a 10 foot shadow. A man who is 6 feet tall casts a 3.5 foot shadow. How tall is
the pole?
x
(29) x = –4.
a) 14.65 b) –16.75 c) 17.79 d) 20.93 e) NOTA
(30) x = 86.
a) 60.93 b) 48.37 c) 32.67 d) 16.97 e) NOTA
For 31 and 32, A is in Quadrant III and sin A =
5
3
.
(31) Find cos A.
a)
5
4
b)
5
4
c)
5
3
d)
5
3
e) NOTA
(32) Find tan A.
a)
4
3
b)
4
3
c)
3
4
d)
3
4
e) NOTA
(33) Angle B is in standard position in Quadrant II, and sin B =
58
3
Find a point on the terminal side of angle B.
a)
3 , 58 b) (3,7) c) (–7,3) d) (7,3) e) NOTA
(34) Angle C is in standard position in quadrant III, and cos C =
9
4
. Find a point on the
terminal side of angle C.
a) 65 , 9 b) 65 , 9 c) 97 , 9 d) 97 , 9 e) NOTA
(35) Find sin
6
7
exactly.
a)
b)
c)
2
1
d)
2
1
e) NOTA
(36) Find cos
3
4
exactly.
a)
b)
c)
2
1
d)
2
1
e) NOTA
(37) Find z exactly.
a) 9 b)
c)
3
18
d) 18 e) NOTA
(38) A function y = f ( x ) is periodic with period 6. If f (2) = 3, find another value for x such that
f ( x ) = 3.
a) 3 b) 5 c) 8 d) 9 e) NOTA
(39) Let m = sin x , with –1 ≤ m ≤ 1. If x = n , with 0 ≤ n ≤
2
, is one solution, find another
solution.
a) – n b) 2π – n c) π – n d) π + n e) NOTA
For 40, 41, 42, and 43 consider the function y = –5 – 4 sin(3 x – 2).
(40) The amplitude is
a) –5 b) –4 c) 3 d) 2 e) NOTA
(41) What is the vertical shift?
a) 5 down b) 5 up c) 4 down d) 4 up e) NOTA
(42) What is the period?
z
r
9
o
Find b.
a) 3.5m b) 3.9m c) 1.9m d) 2.7m e) NOTA
(49) In ∆ ABC
a = 5 ft
b = 3 ft
C = 68°
Find c. Round to two places.
a) 45.24 ft b) 22.7 ft c) 6.73 ft d) 4.77 ft e) NOTA
(50) Solve sin x = .5 on [0, 2π].
a)
6
is the only solution b)
3
is the only solution
c)
6
6
d)
3
e) NOTA
(51) What is the range of y = sin x?
a) [−1, 1] b) [0, 2π] c) [−π, π] d) (−∞,∞) e) NOTA
(52) What is the range of y = sin
−
x?
a) [0, 2π] b) [−1, 1] c) [0, π] d)
2
,
2
e) NOTA
(53) What is the range of y = tan
−
x?
a) (−∞,∞) b) [0, 2π] c) [0, π] d)
2
,
2
e) NOTA
(54) y = g ( x ) is periodic with period 5; g (6.1) = 9.7. Find g (21.1).
a) 5 b) 6.1 c) 9.7 d) 24.7 e) NOTA
(55) A cosine function has period 12 and its maximum value at x = 5. At what x value will the
function have a minimum?
a) 8 b) 11 c) 14 d) 17 e) NOTA
(56) A tangent function has period 16, a horizontal intercept at x = 9, and no vertical shift. At
what x value will the graph of this function have a vertical asymptote?
a) 13 b) 17 c) 21 d) 25 e) NOTA
(3) Find θ)? and x.
(4) Find α and x.
20 cm x
15 cm
o
15 cm x
10 cm
o
y x
o
20 cm
(5) Use the fundamental identities to find the exact value of sin x , csc x , and tan x given that
3
2
cos x and csc x 0.
(6) Use a sketch of the unit circle to explain why:
a) the function y = sin x is periodic.
b) the function y = tan x has the vertical asymptotes where it does
c) the function y = cos x has the range that it does
(7) Use the identity for cos( x + y ) to derive an identity for cos(2 x ).
(8) Find the exact value of
12
sin
(9) Find the exact value of
12
5
cos
For Problems 10, 11, and 12 find one solution analytically, then find other solutions by any
method.
(10) 4.4 − 3.2 cos(1.2 x ) = 5.1 on the interval [0, 10].
(11) 2.7 tan(.2 x ) = −6.5 on [0, 50].
(12) 500 + 25 sin(.52 x ) = 512 on [0, 12].
(13) Angle
3
2
A , and angle A is in standard position. The terminal side of angle A intersects
the unit circle at the point ( a , b ). Find the exact values of a and b.
a)
x 0 1 2 3 4
y 6 4 2 4 6
b)
x 0
y Undefined 0 Undefined 0
c)
x -1 -0.5 0 0.5 1 1.
y -10 -7 -10 -13 -10 -
(19) A sine function y f ( ) x has a period of 8 and amplitude of 3. It is also known that
f (2) 18 is a minimum value of the function.
a) Find the maximum value of the function.
b) List three values for
x that produce a maximum value.
c) List three values for x that product the average value of the function.
d) Write an equation for this function.
(20) Solve each triangle ABC , if possible. If there is no such triangle, explain how you know. If
two triangles are possible, solve both.
a) a = 5 cm, b = 6 cm, c = 7 cm
b) a = 6.17 in, b = 11.52 in, c = 17.41 in
c) a = 5 ft, b = 7 ft, C = 78 deg
d) a = 10 m, b = 6 m, B = 25 deg
e) a = 5 cm, b = 6 cm, B = 35 deg
f) A = 40 deg, B = 50 deg, c = 6 ft
a) On the given axes, make a sketch of y = cos x.
b) Indicate on the graph approximate solutions to cos x = –.
c) For each solution indicate the corresponding quadrant of the unit circle.
(22) Repeat #21 with y = sin x and sin x = –.
(23) A cosine function y g x ( ) has its average value at x 2 and x 10.
