Principles of Math 12 - Trigonometry II Practice Exam
1
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Trigonometry II Practice Exam
Use this sheet to record your answers
1.
NR 2.
19.
28.
2.
NR 3.
20.
NR 7.
NR 1.
11.
21.
29.
3.
12.
NR 5.
30.
4.
13.
22.
31.
5.
14.
23.
32.
6.
NR 4.
24.
33.
7.
15.
25.
8.
16.
NR 6.
9.
17.
26.
10.
18.
27.
Copyright
©Math
Barry
Mabillard,
2006Exam
Principles of
12 - Trigonometry
II Practice
2
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Trigonometry II Practice Exam
1.
The exact value of sin 75˚ can be determined using the expression
A.
B.
C.
D.
sin 90˚ - sin 15˚
sin 45˚ + sin 30˚
sin 45˚cos 30˚ + cos 45˚sin 30˚
cos 45˚cos 30˚ + cos 45˚cos 30˚
Use the following information to answer the next question.
A student solves the equation tan x = −1 in their graphing calculator as shown in
the diagram below.
The student determines the general solution of this graph is −
2.
π
4
+ nπ , n ∈ I
The general solution to the equation tan ( 5 x ) = −1 is
A. −
π
3
π
+
nπ
, n∈ I
10
+ nπ , n ∈ I
4
5π
C. −
+ 5nπ , n ∈ I
4
π nπ
D. − +
, n∈ I
20 5
B. −
Principles of Math 12 - Trigonometry II Practice Exam
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Numerical Response
1.
cos 2 x + sin x
may be verified by substituting 2.1 rad
sin 2 x
for x on each side. When this substitution is made, the numerical value of each
side, to the nearest hundredth, is _________.
The identity cot 2 x + csc x =
Use the following information to answer the next question.
A Ferris wheel at an amusement park, with a diameter of 12 m, can be
π
t + 9 , where h ( t ) is the
20
height above the ground in metres, and t is the time in seconds.
modeled using the equation h(t ) = −6 cos
3.
The number of seconds required for a rider to reach a height of 14 m for the first
time is, to the nearest tenth,
A.
B.
C.
D.
4.
16.3 s
16.5 s
20.4 s
932 s
The expression cot 2 x + csc x − 4 is equivalent to
A. csc 2 x + csc x − 5
B. csc 2 x + csc x − 3
cos 2 x
1
C.
+
−4
2
sin x cos x
D. -4
Principles of Math 12 - Trigonometry II Practice Exam
4
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Use the following information to answer the next question.
The graph of f ( x ) = cos 2 x + cos x + 1 is shown below
5.
If the equation cos 2 x + cos x + 1 = 3 has the general solution 2nπ , n ∈ I ,
⎛ x⎞
⎛ x⎞
then a possible solution to the equation cos 2 ⎜ ⎟ + cos ⎜ ⎟ + 1 = 3 is
⎝3⎠
⎝3⎠
A. 2π
B. 3π
C. 9π
D. 12π
6.
If sin A =
A.
B.
C.
D.
m
m2
and tan A = 3 , where m, n ≠ 0, then cos A is equivalent to
n
n
n2
m
m3
n4
mn 2
1
mn 2
Principles of Math 12 - Trigonometry II Practice Exam
5
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7.
The expression
A.
1 + tan 2 x
, is equivalent to
1 − sin 2 x
(1 + tan x )(1 − tan x )
(1 + sin x )(1 − sin x )
B. 1
C. sec x
D. sec 2 x
8.
If tan 2 x =
5
, then sec 2 x is equivalent to
7
12
7
7
B.
5
5 74
C.
74
74
D.
7
A.
9.
The expression cos 2 ( 4π ) − sin 2 ( 4π ) is equivalent to
A. cos 2 ( 4π )
B. sin 2 ( 8π )
C. cos ( 8π )
D. cos ( 4π ) sin ( 4π )
Principles of Math 12 - Trigonometry II Practice Exam
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Use the following information to answer the next question.
The graphs of f ( x ) = cos x and g ( x ) = cos ( 2 x ) are shown below
The graphs intersect four times on the interval 0 ≤ x ≤ 2π
10.
If the domain is changed to 0 < x < 2π , (the equality has been removed)
a correct statement is
A.
B.
C.
D.
There are more solutions
There are fewer solutions
There are the same number of solutions
There is no change in the number of solutions
Numerical Response
2.
The equation csc 2 x − 2 = cos 2 x has four solutions in the interval 0 < x < 2π .
The number of solutions for x in the interval 0 < x < 14π is _________.
Principles of Math 12 - Trigonometry II Practice Exam
7
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Use the following information to answer the next question.
The steps used by a student to simplify the expression ( sin x + cos x ) are
2
shown below
Step 1 :
sin 2 x + cos 2 x
Step 2 :
sin 2 x + (1 − sin 2 x )
Step 3 :
(1- cos x ) + (1 − sin x )
Step 4 :
2 - sin 2 x − cos 2 x
2
2
Numerical Response
3.
11.
The step which contains a mathematical error is step _______.
The value of m in the equation
m sin x cot x
= 8 is
4 csc x tan x
sin x cos x
32
2sin x cos x
B.
tan x
C. 32 sec 2 x
D. 32sec x csc x
A.
12.
The solutions to the equation cos 2 x = cos x , where 0 ≤ x < 2π are
A. 0,
B.
π
2
,π ,
π 3π
3π
, 2π
2
,
2 2
C. 0,
π 3π
2
,
2
π 3π
D. 0, , , 2π
2 2
Principles of Math 12 - Trigonometry II Practice Exam
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13.
The expression
A.
B.
C.
π
6
π
6
π
± 2nπ ,
5π
± 2nπ
6
± nπ
± 2nπ ,
6
nπ
D.
2
14.
cos x
is undefined when the values of x are
1 − 2sin x
5π
π
± 2nπ , ± nπ
6
2
π⎞
⎛
The expression sec ⎜ x − ⎟ is equivalent to
2⎠
⎝
π
A. sec x − sec
2
π⎞
⎛
B. cos ⎜ x − ⎟
2⎠
⎝
C. csc x
⎛π
⎞
D. − sin ⎜ − x ⎟
⎝2
⎠
Numerical Response
4.
15.
1
π
= 0.43 , and 0 ≤ x < , then the value of x in radians, to the
2
2
1 + cot x
nearest tenth, is ________.
If
The expression
sin x
1
is equivalent to
+
tan x sec x
A. 2 cos x
B. 2sec x
sin x + 1
C.
tan x + sec x
sin x
D.
tan x sec x
Principles of Math 12 - Trigonometry II Practice Exam
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16.
7
4
and cos B = , where A and B are acute angles, the value of
8
5
cos ( A − B ) is equal to
Given sin A =
3
8
16 49
B.
+
25 64
4 15 + 21
C.
40
28 − 3 15
D.
40
A.
17.
If the equation −5 csc 2 x + 12 cot 2 x − 9 = 0 is simplified using the identity
1 + cot 2 x = csc 2 x , the resulting equation is
A.
B.
C.
D.
18.
−5 tan 2 x + 12 sec 2 x − 9 = 0
cot 2 x = 2
12 cot 2 x − 9 − 5sec 2 x = 0
sec 2 x (1 − tan 2 x ) = 6
The expression cos ( x − y ) − cos ( x + y ) is equivalent to
A. 2 sin x sin y
B. 0
C. −2 cos y
⎛ x− y⎞
D. cos ⎜
⎟
⎝ x+ y⎠
19.
1
intersects the graph of cos 2 x − sin x twice in the interval
2
0 ≤ x < 2π . An equation that can be used to solve for x is
The line y =
A.
B.
C.
D.
cos 2 x = sin x
2 cos 2 x − 2 sin x − 1 = 0
sin x − cos 2 x = 2
2 cos 2 x + 2 sin x − 1 = 0
Principles of Math 12 - Trigonometry II Practice Exam
10
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20.
21.
⎛θ ⎞
⎛ 2θ
The expression sin ⎜ ⎟ cos ⎜
⎝5⎠
⎝ 7
⎛ 17θ ⎞
A. cos ⎜
⎟
⎝ 35 ⎠
⎛ 17θ ⎞
B. sin ⎜
⎟
⎝ 35 ⎠
⎛ 3θ ⎞
C. sin ⎜ ⎟
⎝ 35 ⎠
⎛ −3θ ⎞
D. sin ⎜
⎟
⎝ 35 ⎠
If
A.
B.
C.
D.
⎞
⎛ θ ⎞ ⎛ 2θ ⎞
⎟ − cos ⎜ ⎟ sin ⎜ ⎟ is equivalent to
⎠
⎝5⎠ ⎝ 7 ⎠
csc 2 x
= 5 , then the value of x, to the nearest hundredth of a radian is
sec 2 x
1.15 + 3.14n , n ∈ I
0.42 + 3.14n , n ∈ I
0.54 + 1.57 n , n ∈ I
0.21 + 1.57 n , n ∈ I
Use the following information to answer the next question.
A student is given four different trigonometric expressions
I
II
III
IV
1
sec x cos x
9
cot 2 x − csc2 x
2cos 2 x + 2sin 2 x
2 cos x
2 cot x sin x
Numerical Response
5.
If the expressions are simplified are ranked, from smallest to largest, the correct
order is ________.
Principles of Math 12 - Trigonometry II Practice Exam
11
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22.
Given sin x = m , an expression for cos 2x , in terms of m, is
A.
B.
C.
D.
23.
1 − 2m 2
1 − 2m
2m 2 − 1
2m − 1
The expression
A. csc x + sin x
sin x + 1
B.
cos 2 x + 1
sin x + 1
C.
sin 2 x
D. 1
24.
1 + csc x
is equivalent to
sin x
Given x = 450 , an equivalent expression to
cos ( x + y )
is
cos y
⎛x⎞
A. cos ⎜ ⎟ + 1
⎝ y⎠
2
B.
(1 − tan y )
2
C. cos x
2 + 2 cos y
D.
2 cos y
25.
⎛ π ⎞
The exact value of sec ⎜ − ⎟ is
⎝ 12 ⎠
4
A.
2− 6
B.
6− 2
C. −750
11π
D.
12
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Numerical Response
6.
26.
The expression cos x may be written as cos 2 kx − sin 2 kx . The value of k, to the
nearest tenth, is ________.
Using the identity cos 2 x = 1 − sin 2 x , the expression cos 2 x − sin 2 x − 1 + 2 sin x
can be simplified to
A. 2sin x (1 − sin x )
B. sin x (1 − 2sin x )
C. sin 2 x + 2sin x
D. 2sin 2 x + 1
27.
6
2
and sin y = − , the exact value of sec ( x + y ) , given that
7
5
3π
3π
≤ x < 2π ,
≤ y < 2π , is
2
2
If tan x = −
A. 21 − 5
B. 5 − 21
7 21
C.
12 85 − 5
D.
28.
5 85
7 21 − 12
The expression csc x − sin x is equivalent to
1
sin 2 x
B. 1
sin x
C.
cos 2 x
D. cot x cos x
A.
Principles of Math 12 - Trigonometry II Practice Exam
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Numerical Response
7.
29.
The number of solutions in the equation tan 2 x = 1 , where 0 ≤ x < 2π ,
is _________.
The expression
sin x + tan x
is equivalent to
cos x + 1
A. csc 2 x
B. tan x
2 sin x
C.
cos x + 1
D. 2 tan x
30.
The expression csc 4 x − 1 is equivalent to
csc4 x
A.
sec4 x
B. cot 4 x
C. cot 2 x ( csc 2 x + 1)
D. cot 2 x ( sec 2 x + 1)
31.
The general solution to the equation sin 4 x = −
7π nπ
±
24 2
5π nπ
B.
,
±
12 4
7π nπ
C.
,
±
24 2
3π nπ
D.
±
12 4
1
is
2
A.
3π nπ
±
12 4
11π nπ
±
24
2
Principles of Math 12 - Trigonometry II Practice Exam
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32.
The expression sec 2x is undefined when x is the angle
A.
B.
π
4
π
2
C. π
D. 2π
Use the following information to answer the next question.
A student solves the equation cos 2 x − 2 = 0 algebraically, using the steps
shown below
( cos x − 2 )( cos x + 2 ) = 0
33.
cos x − 2 = 0
→ x has no solution.
cos x + 2 = 0
→ x has no solution.
The reason why cos 2 x − 2 = 0 has no solution is because
A.
B.
C.
D.
cos x is undefined for x = 2 ± 2nπ
The range of y = cos x is −1 ≤ y ≤ 1
cos 2 x − 2 = 0 cannot be factored
cos 2 x must be replaced with sin 2 x − 1 before factoring
Principles of Math 12 - Trigonometry II Practice Exam
15
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Use the following information to answer the next question.
A student graphs the following function in a graphing calculator.
f ( x ) = 8 − 3sin 2 x
x is measured in radians, and the student wishes to analyze the graph for −2π ≤ x ≤ 2π
Written Response – 10%
1.
•
Explain how the student would have to type the above equation
into their graphing calculator in order to obtain the correct graph.
Indicate appropriate window settings.
•
The student now wishes to solve the equation 6.2 = f ( x ) .
State the general solution to this equation in radian decimal form,
to the nearest hundredth.
•
The graph of f ( x ) can be expressed in the form
g ( x ) = a cos b [ x − c ] + d . Write the equation for g ( x )
•
Algebraically solve the equation 7 + sin 2 x = 8 − 3sin 2 x
Show all steps required in obtaining the answer.
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Written Response – 10%
2.
•
Verify the identity
cos x
1 + sin x
π
=
for x =
1 − sin x
cos x
6
Use the following additional information to answer the next part of the question.
The graphs of y1 =
•
cos x
1 + sin x
and y2 =
are shown below.
1 − sin x
cos x
The graph of f ( x ) can be expressed in the form
y1 = a sin b [ x − c ] + d or y2 = a cos b [ x − c ] + d .
Write the equations for both y1 and y2.
•
cos x
1 + sin x
and y2 =
are not
1 − sin x
cos x
identical. Explain the difference between the graphs of
y1 and y2.
The graphs of y1 =
Principles of Math 12 - Trigonometry II Practice Exam
17
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•
Algebraically prove the identity
•
Algebraically show that
cos x
1 + sin x
=
1 − sin x
cos x
cos x 1 + sin x 2 cos x
+
=
1 − sin x
cos x
1 − sin x
Principles of Math 12 - Trigonometry II Practice Exam
18
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Written Response – 10%
3.
•
Prove the identity
1 + cos 2 x
= cot x
sin 2 x
•
Prove the identity
(sin x + cos x) 2 = 1 + sin 2 x
•
Prove the identity
sin 2 x = 2 sin x cos x
Principles of Math 12 - Trigonometry II Practice Exam
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• Solve algebraically: 2 sin x cos x = cos x
• Solve algebraically:
sin x sin x
=
2
3
• Solve algebraically:
csc x csc x 16
+
=
5
3
15
Principles of Math 12 - Trigonometry II Practice Exam
20
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