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Solving Trigonometric Equations — Cumulative exam Answers

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Let cos(2x) – cos(x) = 0, where 0° ≤ x ≤ 180°. What are the possible values for x?

A
60° only
B
120° only
C
0° or 120°
D
60° or 180°
22
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What are all the exact solutions of for ? Give your answer in radians.

Question illustration
A
and
Option A
B
, and
Option B
C
, and
Option C
D
, and
Option D
23

Let sin(2x) – sin(x) = 0, where 0 ≤ x < 2π. What are the possible solutions for x?

A
Left-brace StartFraction pi Over 6 EndFraction, StartFraction 5 pi Over 6 EndFraction Right-brace
Option A
B
Left-brace StartFraction pi Over 3 EndFraction, StartFraction 5 pi Over 3 EndFraction right-brace
Option B
C
Left-brace StartFraction pi over 6 EndFraction, StartFraction pi Over 2 EndFraction, StartFraction 5 pi Over 6 EndFraction, StartFraction 3 pi Over 2 EndFraction right-brace
Option C
D
Left-brace 0, StartFraction pi Over 3 EndFraction, pi, StartFraction 5 pi Over 3 EndFraction right-brace
Option D
24

Chris wanted to transform the graph of the parent function by horizontally compressing it so that it has a period of units, horizontally translating it units to the right, and vertically translating it 1 unit up. To do so, he graphed the function , as shown. What did he do wrong?

Question illustration
A
He graphed the function incorrectly because he didn’t translate the graph of the parent function 1 unit up.
Option A
B
He graphed the function incorrectly because the period of the transformed function is not units.
Option B
C
He graphed the function correctly, but it was not the right function to graph. He should have graphed .
Option C
D
He graphed the function correctly, but it was not the right function to graph. He should have graphed .
Option D
25

Which is a solution of ? Give your answer in radians.

Question illustration
A
0.65
B
mc014-2.jpg
Option B
C
mc014-3.jpg
Option C
D
6.15
26

Which is the graph of the function ?

Question illustration
A
On a coordinate plane, the x axis ranges from negative 2 pi to 2 pi with an interval of pi by 2 units and the y axis ranges from negative 6 to 6 with an interval of 2 units. It shows a curve that oscillates between positive and negative infinity as x approaches multiples of pi. The curve has vertical asymptotes at x equals plus or minus n times pi. The graph increases in every pi interval as x increases. The curve crosses the x axis at (2 pi comma 0), (pi comma 0) and (negative pi comma 0) and (negative 2 pi comma 0).
Option A
B
On a coordinate plane, the x axis ranges from negative 2 pi to 2 pi with an interval of pi by 2 units and the y axis ranges from negative 6 to 6 with an interval of 2 units. It shows a curve that oscillates between positive and negative infinity as x approaches multiples of pi. The curve has vertical asymptotes at x equals plus or minus n times pi. The graph decreases in every pi interval as x decreases. The curve crosses the x axis at (2 pi comma 0), (pi comma 0) and (negative pi comma 0) and (negative 2 pi comma 0).
Option B
C
On a coordinate plane, the x axis ranges from negative 2 pi to 2 pi with an interval of pi by 2 units and the y axis ranges from negative 6 to 6 with an interval of 2 units. It shows a curve that oscillates between positive and negative infinity as x approaches multiples of pi. The curve has vertical asymptotes at x equals plus or minus n times pi by 2. The graph decreases in every pi interval as x decreases.The curve crosses the x axis at (2 pi comma 0), (pi comma 0) and (negative pi comma 0) and (negative 2 pi comma 0).
Option C
D
On a coordinate plane, the x axis ranges from negative 2 pi to 2 pi with an interval of pi by 2 units and the y axis ranges from negative 6 to 6 with an interval of 2 units. It shows a curve that oscillates between positive and negative infinity as x approaches multiples of pi. The curve has vertical asymptotes at x equals plus or minus n times pi by 2. The graph increases in every pi interval as x increases. The curve crosses the x axis at (2 pi comma 0), (pi comma 0) and (negative pi comma 0) and (negative 2 pi comma 0).
Option D
27

Which expression is equivalent to ?

Question illustration
A
StartFraction 1 Over cosine (2 x) EndFraction
Option A
B
StartFraction 1 Over sine (2 x) EndFraction
Option B
C
Cosine (2 x)
Option C
D
Sine (2 x)
Option D
28

Review the proof of the identity cos(π − A) = −cosA.cos(π − A)Step 1: = cosπcosA − sinAsinπStep 2: = (−1)(cosA) + (sinA)(0)Step 3: = −1cosA + (sinA)(0)Step 4: = −cosA + 0Step 5: = −cosA

A
step 1
B
step 2
C
step 3
D
step 4
29

Which function is graphed below?

Question illustration
A
mc009-2.jpg
Option A
B
mc009-3.jpg
Option B
C
mc009-4.jpg
Option C
D
mc009-5.jpg
Option D
30

The parent secant function is shifted 2 units down, and its period is changed to . Which of the following is the graph of the transformed function?

Question illustration
A
On a coordinate plane, the x axis ranges from negative pi to pi with an interval of pi by 2 units and the y axis ranges from negative 2 to 4 with an interval of 1 unit. A secant function with asymptotes at x equals plus or minus n pi is drawn. It touches the line y equals 3 at (0 comma 3). The graph touches another line y equals 1 at (pi comma 1) and (negative pi comma 1).
Option A
B
On a coordinate plane, the x axis ranges from negative 2 pi to 2 pi with an interval of pi units and the y axis ranges from negative 4 to 2 with an interval of 1 unit. A secant function with asymptotes at x equals plus or minus n pi is drawn. It touches the line y equals negative 3 at (2 pi comma negative 1),(0 comma negative 1) and (negative 2 pi comma negative 1). The graph touches another line y equals negative 3 at (pi comma negative 3) and (negative pi comma negative 3).
Option B
C
On a coordinate plane, the x axis ranges from negative pi to pi with an interval of pi units and the y axis ranges from negative 2 to 4 with an interval of 1 unit. A secant function with asymptotes at x equals plus or minus n pi is drawn. It touches the line y equals 3 at (pi comma 3), (0 comma 3) and (negative pi comma 3). The graph touches another line y equals 1 at (pi by 2 comma 1) and (negative pi by 2 comma 1).
Option C
D
On a coordinate plane, the x axis ranges from negative pi to pi with an interval of pi units and the y axis ranges from negative 4 to 2 with an interval of 1 unit. A secant function with asymptotes at x equals plus or minus n pi is drawn. It touches the line y equals negative 1 at (pi comma negative1), (0 comma negative1) and (negative pi comma negative 1). The graph touches another line y equals negative 3 at (pi by 2 comma negative 3) and (negative pi by 2 comma negative 3).
Option D
31

Consider the derivation of an alternate form of the cosine double angle identity.

Question illustration
A
In step 1, cos(2x) is equal to cos2(x) + sin2(x).
B
In step 2, sin2(x) should have been replaced with 1 + cos2(x).
C
In step 3, cos2(x) – 1 – cos2(x) should be cos2(x) – 1 + cos2(x).
D
In step 4, 2cos2(x) – 1 should be 1 – 2cos2(x).
32

Which expression is equivalent to cos(40°)cos(10°) + sin(40°)sin(10°)?

A
sin(40° – 10°)
B
cos(40° + 10°)
C
sin(40° + 10°)
D
cos(40° – 10°)
33

What is the range of y = sin-1x?

A
[–1, 1]
B
[0, π]
C
Left-bracket negative StartFraction pi Over 2 EndFraction, StartFraction pi Over 2 EndFraction right-bracket
Option C
D
(–∞, ∞)
34

How is the graph of y = csc(x – 6) transformed from its parent function?

A
It is the graph of y = csc(x) shifted 6 units up.
B
It is the graph of y = csc(x) shifted 6 units right.
C
It is the graph of y = csc(x) shifted 6 units left.
D
It is the graph of y = csc(x) shifted 6 units down.
35

Let tan(2x) + tan(x) = 0, where 0 ≤ x < 2π. How many possible solutions are there for x?

A
2
B
4
C
6
D
8
36

Which expression is equivalent to cos(4x)?

A
2(2cos2(x) – 1)2 – 1
B
2cos2(x) – 1
C
2cos2(x) – 4
D
4cos2(x)
37

Which expression is equivalent to 4sin2(x) – 8sin2?

Question illustration
A
4(cos(x) – cos2(x))
B
4(–cos(x) – cos2(x))
C
4(2 – cos(x) – cos2(x))
D
4(2 + cos(x) – cos2(x))
38

Review the graph.

Question illustration
A
y = sinx
B
y = cosx
C
y = arcsinx
D
y = arccosx
39

Which of the following is an asymptote of y = sec(x)?

A
x equal to minus 2 pi.
Option A
B
x equal to minus StartFraction pi over 6 EndFraction.
Option B
C
x equal to pi.
Option C
D
x equal to StartFraction 2 pi over 2 EndFraction.
Option D
40

Which set of steps can be used to prove the sine sum identity, sin(x + y) = sin(x)cos(y) + cos(x)sin(y)?

A
Use the complementary relationship between sine and cosine to rewrite sin(x + y) as . Apply the cosine sum identity. Then simplify using sin(–y) = –sin(y) and cos(–y) = cos(y).
Option A
B
Use the complementary relationship between sine and cosine to rewrite sin(x + y) as . Apply the cosine sum identity. Then simplify using sin(–y) = sin(y) and cos(–y) = –cos(y).
Option B
C
Use the supplementary relationship between sine and cosine to rewrite sin(x + y) as . Apply the cosine sum identity. Then simplify using sin(–y) = –sin(y) and cos(–y) = cos(y).
Option C
D
Use the supplementary relationship between sine and cosine to rewrite sin(x + y) as . Apply the cosine sum identity. Then simplify using sin(–y) = sin(y) and cos(–y) = –cos(y).
Option D

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