AnswersGA-GSE Pre-CalculusInverse of a Function

Trigonometric Half Angle Identities Answers

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1
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For 0 ≤ ϴ < 2π , what are the solutions to sin2(ϴ) = 2sin2 ?

Question illustration
A
Left-brace 0, StartFraction pi Over 2 EndFraction, StartFraction 3 pi Over 2 EndFraction right-brace
Option A
B
Left-brace 0, StartFraction pi Over 2 EndFraction, Pi right-brace
Option B
C
Left-brace 0, pi, StartFraction 3 pi Over 2 EndFraction right-brace
Option C
D
Left-brace StartFraction pi over 2 EndFraction, pi, StartFraction 3 pi Over 2 EndFraction right-brace
Option D
2
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Which expression is equivalent to ?

Question illustration
A
– 1 – cos(ϴ)
B
– 1 + cos(ϴ)
C
1 + cos(ϴ)
D
1 – cos(ϴ)
3

What is the exact value of sin(112.5°)?

A
StartFraction 2 over StartStartRoot 2 minus StartRoot 2 EndRoot EndEndRoot EndFraction
Option A
B
StartFraction 2 over StartStartRoot 2 + StartRoot 2 EndRoot EndEndRoot EndFraction
Option B
C
StartFraction StartStartRoot 2 minus StartRoot 2 EndRoot EndEndRoot Over 2 EndFraction
Option C
D
StartFraction StartStartRoot 2 + StartRoot 2 EndRoot EndEndRoot Over 2 EndFraction
Option D
4

What is the exact value of tan(67.5°)?

A
StartRoot 3 EndRoot
Option A
B
StartFraction StartRoot 3 EndRoot Over 3 EndFraction
Option B
C
StartStartRoot StartFraction 2 minus StartRoot 2 EndRoot Over 2 + StartRoot 2 EndRoot EndFraction EndEndRoot
Option C
D
StartStartRoot StartFraction 2 + StartRoot 2 EndRoot Over 2 minus StartRoot 2 EndRoot EndFraction EndEndRoot
Option D
5

What is the exact value of ?

Question illustration
A
Negative StartFraction StartStartRoot 2 + StartRoot 3 EndRoot EndEndRoot Over 2 EndFraction
Option A
B
Negative StartFraction StartStartRoot 2 minus StartRoot 3 EndRoot EndEndRoot Over 2 EndFraction
Option B
C
StartFraction StartStartRoot 2 minus StartRoot 3 EndRoot EndEndRoot Over 2 EndFraction
Option C
D
StartFraction StartStartRoot 2 + StartRoot 3 EndRoot EndEndRoot Over 2 EndFraction
Option D
6

Which expression is equivalent to 2cos2?

Question illustration
A
0
B
1
C
1 – 2cos(x)
D
1 + 2cos(x)
7

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))
8

Review the proof of .

Question illustration
A
step 2
B
step 4
C
step 5
D
step 8
9

For 0 ≤ ϴ < 2π, how many solutions are there to tan = sin(ϴ)? Note: Do not include values that are undefined for tan or sin(ϴ).

Question illustration
A
0
B
1
C
2
D
3
10

What is the exact value of sin(75°)?

A
One-half
Option A
B
StartFraction StartRoot 3 EndRoot Over 2 EndFraction
Option B
C
StartFraction StartStartRoot 2 minus StartRoot 3 EndRoot EndEndRoot Over 2 EndFraction
Option C
D
StartFraction StartStartRoot 2 + StartRoot 3 EndRoot EndEndRoot Over 2 EndFraction
Option D
11

Consider the derivation of sin(2x).

Question illustration
A
Step 2 should read = cos(x)cos(x) – sin(x)sin(x).
B
Step 2 should read = sin(x)cos(x) – cos(x)sin(x).
C
Step 3 should read = – 2sin(x)cos(x).
D
Step 3 should read = 2sin(x)cos(x).
12

Suppose where . What is the value of tan(2x)?

Question illustration
A
Negative StartFraction 2 StartRoot 8 EndRoot Over 7 EndFraction
Option A
B
Negative StartRoot 8 EndRoot
Option B
C
StartRoot 8 EndRoot
Option C
D
StartFraction 2 StartRoot 8 EndRoot Over 7 EndFraction
Option D
13

Review the proof (not in order) of the identity.

Question illustration
A
1-2-3-4-5
B
1-2-4-3-5
C
1-4-2-3-5
D
1-4-3-2-5
14

Suppose and cos(x) < 0. What is the value of cos(2x)?

Question illustration
A
Negative StartFraction 14 Over 25 EndFraction
Option A
B
Negative StartFraction 7 Over 25 EndFraction
Option B
C
StartFraction 7 Over 25 EndFraction
Option C
D
StartFraction 14 Over 25 EndFraction
Option D
15

Which expression is equivalent to ?

Question illustration
A
cos(5x)
B
cos(20x)
C
sin(5x)
D
sin(20x)
16

Read the proof.

Question illustration
A
sin(2x) = cos(x)cos(x) – sin(x)sin(x)
B
sin(2x) = sin(x)cos(x) – sin(x)cos(x)
C
sin(2x) = cos(x)cos(x) + sin(x)sin(x)
D
sin(2x) = sin(x)cos(x) + sin(x)cos(x)
17

What is the value of 2cos2(75°) – 1?

A
Negative StartFraction StartRoot 3 EndRoot Over 2 EndFraction
Option A
B
Negative one-half
Option B
C
One-half
Option C
D
StartFraction StartRoot 3 EndRoot Over 2 EndFraction
Option D
18

Which expression is equal to ?

Question illustration
A
Negative sine (StartFraction pi Over 20 EndFraction)
Option A
B
Negative sine (StartFraction pi Over 5 EndFraction)
Option B
C
Sine StartFraction pi Over 5 EndFraction
Option C
D
Sine StartFraction pi Over 20 EndFraction
Option D
19

Review the proof.

Question illustration
A
sin2(x)
B
2sin(x)
C
2sin(x)cos(x)
D
sin(x)cos(x) + sin(x)
20

Which derivation correctly uses the cosine sum identity to prove the cosine double angle identity?

A
A 2-column table with 3 rows. Column 1 has entries 1, 2, 3. Column 2 is labeled Step with entries cosine (2 x) = cosine (x + x), = cosine (x) cosine (x) minus sine (x) sine (x), = cosine squared (x) minus sine squared (x).
Option A
B
A 2-column table with 3 rows. Column 1 has entries 1, 2, 3. Column 2 is labeled Step with entries cosine (2 x) = cosine (x + x), = cosine (x) cosine (x) + sine (x) sine (x), = cosine squared (x) + sine squared (x).
Option B
C
A 2-column table with 3 rows. Column 1 has entries 1, 2, 3. Column 2 is labeled Step with entries cosine (2 x) = cosine (x + x), = sine (x) sine (x) + cosine (x) cosine (x), = sine squared (x) + cosine squared (x).
Option C
D
A 2-column table with 3 rows. Column 1 has entries 1, 2, 3. Column 2 is labeled Step with entries cosine (2 x) = cosine (x + x), = sine (x) sine (x) minus cosine (x) cosine (x), = sine squared (x) minus cosine squared (x).
Option D

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