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Evaluating the Six Trigonometric Functions Answers

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1
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Francesca drew point (–2, –10) on the terminal ray of angle , which is in standard position. She found values for the six trigonometric functions using the steps below.Step 1Step 2 Step 3Which of the following explains whether all of Francesca’s work is correct?

Question illustration
A
Each step is correct because she plotted the point, drew a line to the x-axis to form a right triangle, used the Pythagorean theorem to find the hypotenuse, and finally wrote the correct ratios for all six functions.
B
She made her first error in step 1 because she should have drawn the line to the y-axis to form the right triangle.
C
She made her first error in step 2 because she should have used a negative value for r.
D
She made her first error in step 3 because the sine, cosine, and tangent ratios are incorrect, which also results in incorrect cosecant, secant, and tangent functions.
2
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Keisha and David each found the same value for , as shown below, given .Keisha’s SolutionDavid’s SolutionWhose procedure is correct?

Question illustration
A
Keisha’s procedure is correct.
B
David’s procedure is correct.
C
Both procedures are correct.
D
Neither procedure is correct.
3

The point P(x, y) lies on the terminal side of an angle = in standard position. What are the signs of the values of x and y?

Question illustration
A
Both x and y are positive.
B
Both x and y are negative.
C
x is positive, and y is negative.
D
x is negative, and y is positive.
4

The point is the point at which the terminal ray of angle intersects the unit circle. What are the values for the cosine and cotangent functions for angle ?

Question illustration
A
cosine theta = Negative StartFraction StartRoot 2 EndRoot Over 2 EndFraction, cotangent theta = negative 1
Option A
B
cosine theta = StartFraction StartRoot 2 EndRoot Over 2 EndFraction, cotangent theta = 1
Option B
C
cosine theta = StartFraction StartRoot 2 EndRoot Over 2 EndFraction, cotangent theta = negative one-half
Option C
D
cosine theta = Negative StartFraction StartRoot 2 EndRoot Over 2 EndFraction, cotangent theta = one-half
Option D
5

The terminal side of an angle in standard position passes through P(–3, –4). What is the value of ?

Question illustration
A
tangent theta = Negative four-thirds
Option A
B
tangent theta = Negative three-fourths
Option B
C
Tangent theta = three-fourths
Option C
D
Tangent theta = four-thirds
Option D
6

If , which expression is equivalent to ?

Question illustration
A
StartFraction 1 Over negative three-eighths EndFraction
Option A
B
Negative three-eighths + 1
Option B
C
StartRoot 1 + StartFraction (Negative eight-thirds) squared EndRoot
Option C
D
(Negative three-eighths) squared + 1
Option D
7

Given that , what is the value of , for ?

Question illustration
A
Negative StartRoot 2 EndRoot
Option A
B
StartRoot 2 EndRoot
Option B
C
0
D
1
8

The point P(x, y) lies on the terminal side of an angle = –60° in standard position. What are the signs of the values of x and y?

Question illustration
A
Both x and y are positive.
B
Both x and y are negative.
C
x is positive, and y is negative.
D
x is negative, and y is positive.
9

The point P(x, y) lies on the terminal side of an angle = in standard position. What are the signs of the values of x and y?

Question illustration
A
Both x and y are positive.
B
Both x and y are negative.
C
x is positive, and y is negative.
D
x is negative, and y is positive.
10

Based on Pythagorean identities, which equation is true?

A
sine squared theta minus 1 = cosine squared theta
Option A
B
secant squared theta minus tangent squared theta = negative 1
Option B
C
negative cosine squared theta minus 1 = negative sine squared theta
Option C
D
cotangent squared theta minus cosecant squared theta = negative 1
Option D
11

Which Pythagorean identity is correct?

A
sine squared (theta) + 1 = cosecant squared (theta)
Option A
B
tangent squared (theta) + 1 = secant squared (theta)
Option B
C
1 minus cotangent squared (theta) = cosecant squared (theta)
Option C
D
1 minus cosine squared (theta) = tangent squared (theta)
Option D
12

Given that , what is the value of , for ?

Question illustration
A
Negative StartFraction 35 Over 12 EndFraction
Option A
B
Negative StartFraction 12 Over 35 EndFraction
Option B
C
StartFraction 12 Over 35 EndFraction
Option C
D
StartFraction 35 Over 12 EndFraction
Option D
13

Francesca drew point (–2, –10) on the terminal ray of angle , which is in standard position. She found values for the six trigonometric functions using the steps below.Step 1Step 2 Step 3Which of the following explains whether all of Francesca’s work is correct?

Question illustration
A
Each step is correct because she plotted the point, drew a line to the x-axis to form a right triangle, used the Pythagorean theorem to find the hypotenuse, and finally wrote the correct ratios for all six functions.
B
She made her first error in step 1 because she should have drawn the line to the y-axis to form the right triangle.
C
She made her first error in step 2 because she should have used a negative value for r.
D
She made her first error in step 3 because the sine, cosine, and tangent ratios are incorrect, which also results in incorrect cosecant, secant, and tangent functions.
14

The terminal side of an angle in standard position passes through P(15, –8). What is the value of ?

Question illustration
A
sine theta = fifteen-seventeenths
Option A
B
sine theta = negative eight-seventeenths
Option B
C
sine theta = eight-seventeenths
Option C
D
sine theta = fifteen-seventeenths
Option D
15

Three trigonometric functions for a given angle are shown below.What are the coordinates of point (x, y) on the terminal ray of angle , assuming that the values above were not simplified?

Question illustration
A
(–77, –36)
B
(–77, 36)
C
(–36, 77)
D
(36, –77)
16

Keisha and David each found the same value for , as shown below, given .Keisha’s SolutionDavid’s SolutionWhose procedure is correct?

Question illustration
A
Keisha’s procedure is correct.
B
David’s procedure is correct.
C
Both procedures are correct.
D
Neither procedure is correct.
17

The point is the point at which the terminal ray of angle intersects the unit circle. What are the values for the cosine and cotangent functions for angle ?

Question illustration
A
cosine theta = Negative StartFraction StartRoot 2 EndRoot Over 2 EndFraction, cotangent theta = negative 1
Option A
B
cosine theta = StartFraction StartRoot 2 EndRoot Over 2 EndFraction, cotangent theta = 1
Option B
C
cosine theta = StartFraction StartRoot 2 EndRoot Over 2 EndFraction, cotangent theta = negative one-half
Option C
D
cosine theta = Negative StartFraction StartRoot 2 EndRoot Over 2 EndFraction, cotangent theta = one-half
Option D
18

If , which of the following represents an approximate value of , for ?

Question illustration
A
0.7660
B
0.8392
C
1.1916
D
1.4198
19

Given that , what is the value of , for ?

Question illustration
A
Negative StartRoot StartFraction 20 Over 29 EndFraction EndRoot
Option A
B
Negative StartFraction 20 Over 29 EndFraction
Option B
C
StartFraction 20 Over 29 EndFraction
Option C
D
StartRoot StartFraction 20 Over 29 EndFraction EndRoot
Option D

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