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Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the fundamental identities to find the value of the trigonometric function.
5
and tan > 0 .
1) Find cos if sin = 13
A) -
12
13
B) -
5
12
C)
Use basic identities to simplify the expression.
2) cot sec sin
A) 1
B) tan2
12
5
1)
D) -
C) sec2
13
5
2)
D) csc2
Simplify the expression.
cos
3)
2
- x tan x
3)
sin 2 x
A) sin 2 x
4)
B) tan x
C) sec x
D) csc x
cos x
cos x
tan 2 x sin 2 x
A) cos x
4)
B) -cos x
C) sin x
D) -sin x
Write each expression in factored form as an algebraic expression of a single trigonometric function.
5) cos x - sin 2 x - 1
A) cos 2 x - 1
B) (cos x + 2)(cos x - 1)
D) sin 2 x
C) (cot x + 1)(cot x - 1)
Find all solutions in the interval [0, 2 ).
6) 2 sin2x = sin x
5
A) x = 0, , ,
6 6
C) x =
2
,
B) x =
3
2
, ,
2 3 3
D) x =
3
6
,
2
3
,
5
6
7) cos2x + 2 cos x + 1 = 0
A) x = 2
5)
C) x =
B) x =
2
,
3
2
6)
D) x =
4
,
7)
7
4
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Prove the identity.
8) (sin x)(tan x cos x - cot x cos x) = 1 - 2 cos 2 x
8)
1
9)
10)
cot x
csc x - 1
=
1 + csc x
cot x
9)
tan t
1 + cot t tan2 t + 2 - csc2 t
+
=
1 - cot t
tan t
tan t - 1
10)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Complete the identity.
11) The expression
sec x + csc x
tan x + cot x
11)
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) cot x + sin2x
B) cos x + 1
C) tan x - sec x
D) sin x + cos x
Find an exact value.
12) cos 15°
- 2+
A)
4
13) sin
12)
6
- 6+1
B)
4
- 2C)
4
6
D)
6+
4
2
11
12
A)
13)
64
2
B)
-
64
2
C)
24
Write the expression as the sine, cosine, or tangent of an angle.
14) sin 32° cos 18° - cos 32° sin 18°
A) sin 14°
B) cos 14°
C) sin 50°
6
D)
6+
4
2
14)
D) cos 50°
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Prove the identity.
15) cos (x - y) - cos (x + y) = 2 sin x sin y
15)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find all solutions to the equation in the interval [0, 2 ).
16) cos x - cot x = 0
3
3 5 7
,
,
A) ,
B) ,
2 2
4 4 4 4
16)
C) 0,
Rewrite with only sin x and cos x.
17) sin 2x - cos 3x
A) 3 sin2 x cos x - sin3 x + 2 sin x cos x
B) cos3 x + 2 sin2 x cos x - sin2 x + 2 sin x cos x
D) No solution
17)
C) 2 sin x cos x + cos x - 4 cos x sin2 x
D) 2 sin2 x cos x - cos3 x - 2 sin x cos x
2
Find the exact value by using a half-angle identity.
18) cos A)
18)
8
1
2
2-
2
B)
1
2
2+
2
C)
1
2
1-
2
D)
1
2
1+
2
Solve the triangle.
19)
A) B = 60°, b
C) B = 45°, b
19)
8.5, c
7.7, c
10.5
8.5
B) B = 30°, b
D) B = 45°, b
9.7, c
8.5, c
10.5
7.7
Two triangles can be formed using the given measurements. Solve both triangles.
20) A = 59°, a = 14, b = 15
A) B = 66.7°, C = 54.3°, c = 14.8; B = 113.3°, C = 7.7°, c = 14.8
B) B = 23.3°, C = 97.7°, c = 16.2; B = 156.7°, C = 82.3°, c = 16.2
C) B = 23.3°, C = 97.7°, c = 12.1; B = 156.7°, C = 82.3°, c = 12.1
D) B = 66.7°, C = 54.3°, c = 13.3; B = 113.3°, C = 7.7°, c = 2.2
20)
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.
21) C = 42°, a = 18, c = 10
21)
A) The triangle cannot be solved with the Law of Sines.
B) A = 79°, B = 59°, b 14.7
C) A = 59°, B = 79°, b 14.7
D) No triangle is formed.
Solve.
22) Two tracking stations are on the equator 124 miles apart. A weather balloon is located on a bearing
of N 34°E from the western station and on a bearing of N 16°E from the eastern station. How far is
the balloon from the western station?
A) 298 miles
B) 395 miles
C) 386 miles
D) 307 miles
3
22)
Solve the triangle.
23)
23)
A) B = 20.3 , C = 34.7 , a = 21.5
C) B = 34.7 , C = 20.3 , a = 21.5
B) B = 20.3 , C = 34.7 , a = 461
D) B = 34.7 , C = 20.3 , a = 461
Find the area. Round your answer to the nearest hundredth if necessary.
24) Find the area of the triangle with the following measurements:
A = 50°, b = 25 ft, c = 14 ft
A) 268.12 ft2
B) 134.06 ft2
C) 112.49 ft2
24)
D) 350 ft2
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.
25) a = 240
25)
b = 125
c = 255
A) 16,779.4
B) 14,859.26
C) 16,765.3
D) No triangle is formed.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
26) A student claims that the equation tan + 1 = sec is an identity, since by letting
get 0 + 1 = 1, a true statement. Comment on this student's reasoning.
= 0° we
27) Explain why the law of sines cannot be used to solve a triangle if we are given the lengths
of two sides and the measure of the included angle.
4
26)
27)