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Chp 5: Analytic Trigonometry PRACTICE Test
Name_________________________________
Date_________________________
Per _________
*****YOU CAN ONLY USE A CALCULATOR FOR #1-10 ONLY!*****
MULTIPLE CHOICE. Choose the best answer for each problem. Show ALL work for full
points.
Use the fundamental identities to find the value of the trigonometric function.
1)
Find tan θ if cos θ =
A) -
and sin θ < 0.
B) -
C)
D) 4
-
2)
Find sin θ if tan θ = A)
and cos θ > 0.
B)
-
C)
D)
-
-
Use basic identities to simplify the expression.
3) cos θ - cos θ sin2θ
A) sec2θ
4) tan2θ csc2θ
A) cos3θ
Simplify the expression.
5) csc (- x) sin (- x)
A) - cot x
6)
(x) A)
B) cos3θ
C) tan2θ
D) sin θ
B) sec2θ
C) sin θ
D) tan2θ
B) sec x
C) 1
D) -1
C) -1
D) 1
(-x)
x
B) -
x
7)
A) -sin x
B) -cos x
C) sin 2x
D) cos2x
A) sin y
B) -sin y
C) csc y
D) cos y
A) cos x
B) sec x
C) cot x
D) sin x
+
A) 2 csc x
B) 2
8)
9)
10)
11)
x
C) 2
x
D)
x
+
A) 2
B) 2 sec x
C)
x
D) 2 csc x
Write each expression in factored form as an algebraic expression of a single trigonometric
function.
12) cos x x-1
A) (cot x + 1)(cot x - 1)
C)
x
B) (cos x + 2)(cos x - 1)
D)
x-1
13) sec4x + sec2x tan2x - 2 tan4x
A) 3 sec2x - 2
B) sec2x + 2
C) tan2x - 1
D) 4 sec2x
Find all solutions in the interval [0, 2π).
14) sin2x - cos2x = 0
A)
B)
x=
,
,
,
C)
x=
,
x=
,
D)
x=
15) 4
x - 4 sin x + 1 = 0
A)
B)
,
C)
,
D)
,
Find all solutions to the equation.
16)
sin x =
A)
B)
C)
D)
17) cos x = sin x
A)
B)
C)
D)
(Express your answer in radians, in exact form.)
,
SHORT ANSWER. Show ALL work for full points and box your answer.
Prove the identity.
18) 1 + sec2x sin2x = sec2x
19)
=
20) 21)
x+
x=
x-
x=
x
x
x+
x
22)
=
23)
x
x = sin x (
x-
x)
24)
25)
+
=
x = tan x (
x-2
x
x + 1)
MULTIPLE CHOICE. Choose the best answer for each problem. Show ALL work for full
points.
Complete the identity.
26)
The expression
is to be the left hand side of an equation that is an identity. Which one of the
following four expressions can be used as the right hand side of the equation to
complete the identity?
A) sin x + cos x
B) cos x + 1
C) cot x + sin2x
D) tan x - sec x
27)
The expression
∙ csc θ
is to be the left hand side of an equation that is an identity. Which one of the
following four expressions can be used as the right hand side of the equation to
complete the identity?
A) csc θ - tan θ
B) sec θ + cos θ
C) tan2θ
D) cos θ - 1
Write the expression as the sine, cosine, or tangent of an angle.
28) cos 133° cos 58° + sin 133° sin 58°
A) sin 75°
B) sin 191°
C) cos 191°
D) cos 75°
29)
sin
A)
cos
sin
30)
+ cos
sin
B)
C)
cos
D)
sin
cos
A)
B)
tan
C)
tan
D)
tan
tan
SHORT ANSWER. Show ALL work for full points and box your answer.
Prove the identity.
31) sin 4x = 2 sin 2x cos 2x
32) cos 3x =
x-3
x cos x
MULTIPLE CHOICE. Choose the best answer for each problem. Show ALL work for full
points.
Find all solutions to the equation in the interval [0, 2π).
33) cos x - cot x = 0
A) 0, π
B)
,
,
,
D) No solution
C)
,
34) 2 cos x + sin 2x = 0
A)
B)
,
C)
0,
, π,
D)
0,
,
,
0, π,
Rewrite with only sin x and cos x.
35) sin 2x - cos 3x
A) 2
x cos x -
x - 2 sin x cos x
B) 3
x cos x x + 2 sin x cos x
C) 2 sin x cos x + cos x - 4 cos x
x
D)
x+2
x cos x -
x + 2 sin x cos x
Find the exact value by using a half-angle identity.
36)
sin
A)
37) tan 75°
A) 2 +
B)
C)
D)
B) -2 +
C) 2 -
D) -2 -
SHORT ANSWER. Show ALL work for full points and box your answer.
Prove the identity.
38)
x=
( 3 + 4 cos 2x + cos 4x)
MULTIPLE CHOICE. Choose the best answer for each problem. Show ALL work for full
points.
Find all solutions in the interval [0, 2π).
39)
x
=
A)
B)
0,
,
C)
0,
D) 0, π
,
40)
sin
A)
=2
x-1
B)
, π,
C)
,
Solve the triangle.
41) A = 40°, B = 29°, b = 11
A) C = 111°, a ≈ 14.5, c ≈ 21
C) C = 111°, a ≈ 8.3, c ≈ 15.9
D) 0, π
, π,
B) C = 21°, a ≈ 8.3, c ≈ 15.9
D) C = 111°, a ≈ 8.3, c ≈ 21
State whether the given measurements determine zero, one, or two triangles.
42) B = 85°, b = 24, c = 21
A) One
B) Two
C) Zero
Two triangles can be formed using the given measurements. Solve both triangles.
43) C = 72°, a = 27, c = 26
A) A = 9°, B = 99°, b = 27; A = 171°, B = 81°, b = 27
B) A = 9°, B = 99°, b = 25; A = 171°, B = 81°, b = 25
C) A = 81°, B = 27°, b = 54.5; A = 99°, B = 9°, b = 54.5
D) A = 81°, B = 27°, b = 12.4; A = 99°, B = 9°, b = 4.3
The given measurements may or may not determine a triangle. If not, then state that no
triangle is formed. If a triangle is formed, then use the Law of Sines to solve the triangle, if it
is possible, or state that the Law of Sines cannot be used.
44) B = 138°, c = 10, b = 12
A) C = 33.9°, A = 8.1°,
B) C = 8.1°, A = 33.9°,
C) No triangle is formed.
D) The triangle cannot be solved with the Law of Sines.
Solve.
45) Two tracking stations are on the equator 158 miles apart. A weather balloon is
located on a bearing of N 41°E from the western station and on a bearing of N
21°E from the eastern station. How far is the balloon from the western station?
A) 431 miles
B) 399 miles
C) 440 miles
D) 408 miles
Solve the triangle.
46) b = 22, c = 29, A = 80°
A) a ≈ 33.2, B ≈ 40.7°, C ≈ 59.3°
C) a ≈ 36.3, B ≈ 44.7°, C ≈ 55.3°
47) a = 7, b = 13, c = 4
A) No triangles possible
C) A ≈ 67.6°, B ≈ 78.6°, C ≈ 13°
B) No triangles possible
D) a ≈ 36.3, B ≈ 40.7°, C ≈ 59.3°
B) A ≈ 33.8°, B ≈ 67.6°, C ≈ 78.6°
D) A ≈ 33.8°, B ≈ 45.9°, C ≈ 100.3°
Find the area. Round your answer to the nearest hundredth if necessary.
48) Find the area of the triangle with the following measurements:
A = 58°, b = 10 m, c = 21 m
A) 89.05
B) 55.64
C) 105
D) 178.09
49) Find the area of a regular decagon (10 sides) inscribed in a circle of radius 16
inches.
A) 1280 cos 36° ≈ 1035.54 square inches
B) 1280 sin 45° ≈ 905.1 square inches
C) 2560 sin 36° ≈ 1504.73 square inches
D) 1280 sin 36° ≈ 752.37 square inches
Decide whether a triangle can be formed with the given side lengths. If so, use Heron's
formula to find the area of the triangle.
50) a = 59.3
b = 65.4
c = 56.2
A) 1554.77
B) No triangle is formed.
C) 1498.64
D) 1484.54
Solve the problem.
51) A ship travels 98 km on a bearing of 38°, and then travels on a bearing of 128°
for 169 km. Find the distance of the end of the trip from the starting point, to the
nearest kilometer.
A) 195 km
B) 267 km
C) 60 km
D) 77 km
52) A parallelogram has sides of length
and
If the longer diagonal has
a length of
what is the angle opposite this diagonal? Give your answer to
the nearest tenth of a degree.
A) 20.7°
B) 53.8°
C) 56.4°
D) 79.9°
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
B
B
B
B
C
C
A
A
A
C
B
B
A
A
C
A
A
1+
x
x=1+
=1+
x=
x.
19)
=
= csc x +1 =
=
.
20)
x+
21)
x=-
x-
+
x=(
(
=
x+
x)(
x-
x + 1) =
x) = (
x
x+
x.
x)(1) =
x+
x.
22)
=
23)
x
=
x = sin x (1 -
x) (
.
=
x) = sin x (
x-
x).
24)
+
25)
26)
27)
28)
29)
30)
31)
32)
33)
34)
35)
36)
37)
38)
=
=
=
x.
x = tan x
= tan x (
x-2
x + 1).
A
A
D
C
A
sin 4x = sin 2x cos 2x + cos 2x sin 2x = 2 sin 2x cos 2x.
cos 3x = cos (2x + x) = cos 2x cos x - sin 2x sin x = (
=
C
A
C
A
A
x-
x=
x cos x - 2
x
x=
x cos x =
=
x-
x-3
x) cos x - 2 sin x cos x sin x
x cos x.
=
(1+ 2 cos 2x +
2x) =
=
39)
40)
41)
42)
43)
44)
45)
46)
47)
48)
49)
50)
51)
52)
A
C
A
A
D
A
A
A
A
A
D
A
A
D
( 3 + 4 cos 2x + cos 4x).