Trigonometry Review Problems
The following problems are provided as examples of some of the trigonometry concepts and
topics that are tested on the mathematics placement test used at UNCW. They are not intended
to be a complete summary of all topics covered on the test or representative of prerequisite
material for any UNCW mathematics course. See our website (www.uncw.edu/mathlab) for
more information about the placement test.
1. State the value of each of the following:
( )
e.) sin (− 225 )
a.) sin 30 o
o
b.) cos(π6 )
f.) csc( 23π )
(
c.) tan 135 o
g.) cot (0 )
)
d.) sec(π )
h.) cos(176π )
2. If θ is an acute angle and sin (θ ) = 23 , find the value of each of the following:
a.) cos(θ )
3. If sin (θ ) =
b.) tan (θ )
1
2
a.) sin (2θ )
and cos(θ ) =
− 3
2
c.) sec(θ )
d.) csc(θ )
, find the value of each of the following:
b.) cos(2θ )
c.) tan (θ )
4. In triangle ABC, the measures of angles A, B, and C are 20 o , 70 o , and 90 o , respectively. If
BC is 8 units long, how long is AC ?
5. In triangle ABC, angle C is a right angle. Let the measure of angle A be α and the measure
of angle B be β . Let the side opposite angle A have length a, the side opposite angle B have
length b, and the hypotenuse have length c. Find each of the following:
a.) cos(α )
6.
b.) tan (α ) tan (β )
c.) sin (α + β )
d.) sin (2α )
1
1
−
equals
1 − sin( x) 1 + sin( x)
7. In triangle ABC, angle at C is a right angle. The measure of angle A is α , and the length of
side AC is 1. If 45 o ≤ α ≤ 60 o , then what are the possible lengths of side BC ?
8. If tan (θ ) =
3
4
and π ≤ θ ≤ 2π , then what is the value of cos(θ ) ?
9. If 2 sin 2 (θ ) + 3 sin (θ ) + 1 = 0 and 0 ≤ θ < 2π , then what are the possible values of θ ?
10. If
sin (θ ) = 1 and 0 ≤ θ < 2π , then what are the possible values of θ ?
11. In right triangle ABC (with right angle at C), the hypotenuse has length 5. The side opposite
angle A has length 2. If the measure of angle A is α , what is sec(α ) ?
12. What is the period of the graph of the function f ( x) = 13 sin (2 x ) ?
13. The lengths of the two legs of a right triangle are 3 cos(θ ) and 3 sin (θ ) . How long is the
hypotenuse?
14. What is the angle of elevation to the top of a building 200 feet tall from a point 12 feet from
the base of the building?
15. What is the exact value of 3 sin (12π ) ?
ANSWERS
1. a.)
(
3
2
b.)
c.)
d.)
e.)
−1
−1
2
2
3
f.)
2
g.) undefined
h.) − 23
2. a.)
5
3
b.)
2 5
5
c.)
d.)
3 5
5
3
2
3. a.) −
b.) 12
3
2
c.) −
3
3
4.
5.
( )
( )
8 tan 70 o or
8 / tan 20 o
a.) bc
b.) 1
)
c.) a 2 + b 2 / c 2
d.) 2ab / c 2
1
2
6.
2 sin( x)
cos 2 ( x)
7.
1 ≤ BC ≤ 3
8.
− 54
9.
7π
6
10.
π
11.
5 21
21
,
11π
6
, or
3π
2
2
12. π
13. 3
14. θ = arctan( 503 )
15.
3
2
2− 3