Download Alg2 - CH13 Practice Test

Alg2 - CH13 Practice Test
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. Find the value of the sine, cosine, and tangent functions for θ where A = 96, B = 28, and C = 100.
____
a. sin θ = 24 ; cos θ = 7 ; tan θ = 24
b. sin θ = 24
25 ; cos θ = 725; tan θ = 77
c. sin θ = 25
7 ; cos θ = 24
7
25 ; tan θ = 24
cos
θ
=
tan
θ
=
d. sin θ = 25
;
;
7
24
24
25
25
25 to find7the value of x.
2. Use a trigonometric
function
____
____
a. x = 50 2
c. x = 50 3
b. x = 45 3
d. x = 25 2
3. After takeoff from an airport, an airplane’s angle of ascent is 10°. The airplane climbs to an
altitude of 10,000 feet. At that point, what is the land distance between the airplane and the
airport? Round your answer to the nearest foot.
a. 1,763 ft
c. 56,713 ft
b. 57,588 ft
d. 15,424 ft
4. Draw an angle with 515° in standard position.
____
____
a.
c.
b.
d.
5. Find the measure of the reference angle for θ = 142°.
a. 128°
c. 38°
b. –38°
d. 218°
6. P(−7,− 2) is a point on the terminal side of θ in standard position. Find the exact value of the six
trigonometric functions for θ.
a.
c.
sin θ = − 2 5353 ; csc θ = − 253 ;
sin θ = 753 ; csc θ = 7 5353 ;
cos θ = − 7 5353 ; sec θ = −
tan θ = 27 ; cot θ = 72
b.
sin θ = −
53
7
; csc θ
53
7
cos θ = 253 ; sec θ =
tan θ = 27 ; cot θ = 72
;
= − 7 5353 ;
d.
cos θ = − 253 ; sec θ = − 2 5353 ;
tan θ = 27 ; cot θ = 72
____
2 53
53
;
sin θ = − 7 5353 ; csc θ = −
53
7
;
cos θ = − 2 5353 ; sec θ = −
tan θ = 72 ; cot θ = 27
53
2
;
7. Use the unit circle to find the exact value of the trigonometric function cos 30°.
a. −1
c. 1
2
b.
3
2
d.
2
2
____
8. Find all possible values of sin−1
3
.
2
a. π
+ ( 2π )n, 2π + ( 2π )n
c. π
+ ( 2π )n, 5π + ( 2π )n
b. 0.0151
d. π
+ ( 2π )n, 3π + ( 2π )n
3
3
6
4
____
____
6
4
9. Solve the equation sin θ = 0.3 to the nearest tenth. Use the restrictions 90° < θ < 180°.
a. θ = 17.5°
c. θ = 162.5°
b. θ = 107.5°
d. θ = 197.5°
10. You can use trigonometry to measure the height of a pyramid in Egypt.
[1.] An archaeologist positions himself 260 ft from the base of a pyramid so that his eye level is 5
ft above the ground. If the pyramid is 500 feet in height, what would be the angle of elevation from
the archaeologist to the top of the pyramid?
[2.] The angle of elevation from the eye level of an archaeologist to the top of a pyramid whose
base is 400 feet away is 50°. To the nearest foot, what is the height of the pyramid?
____
____
a. [1.] θ ≈ 62°
c. [1.] θ ≈ 62°
[2.] h ≈ 482 ft
[2.] h ≈ 336 ft
b. [1.] θ ≈ 63°
d. [1.] θ ≈ 63°
[2.] h ≈ 477 ft
[2.] h ≈ 341 ft
11. A triangle has a side with length 6 feet and another side with length 8 feet. The angle between the
sides measures 73º. Find the area of the triangle. Round your answer to the nearest tenth.
a. 23.0 ft 2
c. 1752.0 ft 2
b. 45.9 ft 2
d. 7.0 ft 2
12. Solve the triangle. m∠N = 118°, m∠P = 33°, and m = 15. Round to the nearest tenth.
a. m∠M = 29°, n ≈ 8.2, p ≈ 16.9
b. m∠M = 29°, n ≈ 27.3, p ≈ 16.9
Numeric Response
c. m∠M = 29°, n ≈ 8.2, p ≈ 13.4
d. m∠M = 29°, n ≈ 27.3, p ≈ 13.4
13. If tan θ = 68 , what is sinθ?
14. A new kind of computer chip is made in the shape of a triangle. The distance of each side is shown
in the diagram. Find the area of the new chip. Round to the nearest hundredth.
Matching
Match each vocabulary term with its definition.
a. sine
b. cosine
c. Pythagorean Theorem
d. tangent
e. trigonometric function
f. special triangle
g. cotangent
h. secant
i. cosecant
____
____
____
____
____
____
15. in a right triangle, the ratio of the length of the leg opposite the angle to the length of the
hypotenuse
16. the reciprocal of the sine function, or the ratio of the length of the hypotenuse to the length of the
leg opposite the angle in a right triangle
17. the reciprocal of the cosine function, or the ratio of the length of the hypotenuse to the length of
the leg adjacent to the angle in a right triangle
18. in a right triangle, the ratio of the length of the leg adjacent to the angle to the length of the
hypotenuse
19. the reciprocal of the tangent function, or the ratio of the length of the leg adjacent to the angle to
the length of the leg opposite the angle in a right triangle
20. a function whose rule is given by a trigonometric ratio
Match each vocabulary term with its definition.
a. angle of rotation
b. vertex angle
c. initial side
d. vector
e.
f.
g.
h.
____
____
____
____
____
____
standard position
terminal side
coterminal angle
reference angle
21. for an angle in standard position, the positive acute angle formed by the terminal side of the angle
and the x-axis
22. the initial position of a rotated ray
23. the terminal position of a rotated ray
24. an angle formed by a rotating ray and a stationary reference ray
25. an angle in standard position with the same terminal side
26. an angle whose vertex is at the origin and whose initial side is on the x-axis
Alg2 - CH13 Practice Test
Answer Section
MULTIPLE CHOICE
1. ANS: C
sin θ =
opp. 28
adj.
opp. 28
=
= 96 = 24 ; tan θ =
=
= 7 ; cos θ =
=
hyp. 100
hyp. 100
adj. 96 7
25
25
24
Feedback
A
B
C
D
The sine is the ratio of the length of the opposite leg to the length of the
hypotenuse.
The cosine is the ratio of the length of the adjacent leg to the length of the
hypotenuse.
Correct!
The tangent is the ratio of the length of the opposite leg to the length of the
adjacent leg.
PTS:
OBJ:
TOP:
2. ANS:
1
DIF: Basic
REF: Page 929
13-1.1 Finding Trigonometric Ratios
NAT: 12.2.1.m
13-1 Right-Angle Trigonometry
D
You know the length of the hypotenuse and want to find the
opp.
sin θ =
length of the side opposite the given angle. Use the sine
hyp.
function.
sin 45° = x
50
2
= x
2
50
x = 25 2
Substitute 45° for θ, x for opp., and 50 for hyp.
Substitute
2
for sin 45°.
2
Multiply both sides by 50 to solve for x.
Feedback
A
B
C
D
The sine of angle is the ratio of the length of the opposite leg to the length of the
hypotenuse.
Use the sine function.
Use the sine function.
Correct!
PTS:
OBJ:
TOP:
3. ANS:
1
DIF: Average
REF: Page 930
13-1.2 Finding Side Lengths of Special Right Triangles
13-1 Right-Angle Trigonometry
C
NAT: 12.2.1.m
opp.
adj.
10, 000
tan 10° =
x
tan θ =
x(tan 10°) = 10,000
x=
10, 000 ≈
56,713
tan 10°
Substitute 10° for θ, 10,000 for opp., and x for adj.
Multiply both sides by x.
Divide both sides by tan 10°. Use a calculator to simplify.
The land distance from the airplane to the airport is about 56,713 feet.
Feedback
A
B
C
D
The tangent function is the ratio of the length of the opposite leg to the length of
the adjacent leg, not the length of the adjacent leg to the length of the opposite
leg.
Use the tangent function, not the sine function.
Correct!
Set your graphing calculator to interpret angle values as degrees, not radians.
PTS: 1
DIF: Average
REF: Page 930
OBJ: 13-1.3 Application
NAT: 12.2.1.m
TOP: 13-1 Right-Angle Trigonometry
4. ANS: D
Start with the initial side on the positive x-axis and rotate the terminal side 515° counterclockwise.
Feedback
A
B
C
D
Start with the initial side on the positive x-axis not the negative x-axis.
Start with the initial side on the positive x-axis not the positive y-axis.
Start with the initial side on the positive x-axis and then rotate the angle
counterclockwise, not clockwise.
Correct!
PTS: 1
DIF: Average
REF: Page 936
OBJ: 13-2.1 Drawing Angles in Standard Position
TOP: 13-2 Angles of Rotation
5. ANS: C
The reference angle is the acute angle created by the terminal side of θ and the x-axis.
For example:
When θ = 105°, the reference angle measures 75°.
When θ = −105°, the reference angle also measures 75°.
Feedback
A
B
C
D
The reference angle is the acute angle between the terminal side of the angle and
the x-axis, not the y-axis.
The reference angle is always positive.
Correct!
The reference angle is an acute angle
PTS: 1
DIF: Basic
REF: Page 937
OBJ: 13-2.3 Finding Reference Angles TOP: 13-2 Angles of Rotation
6. ANS: A
Step 1 Plot point P, and use it to sketch angle θ in standard position.
Find r.
r=
(−7)2 + (−2)2 =
53
Step 2 Find sin θ, cos θ, and tan θ.
sin θ =
y
y
= −2 = − 2 53 ; cos θ = x = −7 = − 7 53 ; tan θ = = −2 = 2
r
53
r
53
x −7 7
53
53
Step 3 Use reciprocals to find csc θ, sec θ, and cot θ.
csc θ = 1 = − 53 ; sec θ = 1 = − 53 ; cot θ = 1 = 7
sin θ
2
cos θ
7
tan θ 2
Feedback
A
B
C
D
Correct!
Plot the point P and sketch the angle in standard position.
Plot the point P and sketch the angle in standard position.
Plot the point P and sketch the angle in standard position.
PTS: 1
DIF: Average
REF: Page 938
OBJ: 13-2.4 Finding Values of Trigonometric Functions
7. ANS: B
ÊÁ
Ë
The angle passes through the point ÁÁÁ
cos θ = x
cos 30° =
3
2
,
1
2
TOP: 13-2 Angles of Rotation
ˆ˜
˜˜˜ on the unit circle.
¯
3
2
Feedback
A
B
C
D
The cosine of an angle is equal to the x-value of the point where the angle
intersects the unit circle.
Correct!
The cosine of an angle is equal to the x-value of the point where the angle
intersects the unit circle, not the y-value.
The cosine of an angle is equal to the x-value of the point where the angle
intersects the unit circle.
PTS:
OBJ:
TOP:
8. ANS:
1
DIF: Average
REF: Page 944
13-3.2 Using the Unit Circle to Evaluate Trigonometric Functions
13-3 The Unit Circle
A
Step 1 Find the values between 0 and 2π radians for which sin θ is equal to
3 = sin π
,
2
3
3 = sin 2π
2
3
3
.
2
Use y-coordinates of points on the unit circle.
Step 2 Find the angles that are coterminal with angles measuring π and 2π radians.
π + ( 2π )n,
3
2π + ( 2π )n
3
3
3
Add integer multiples of 2π radians, where n
is an integer.
Feedback
A
B
C
D
Correct!
Find the possible values of the inverse of the sine function, not the reciprocal of
the sine function.
Find the angles on the unit circle where the y-value is the square root of 3 over 2,
then add integer multiples of 2 pi.
Find the angles on the unit circle where the y-value is the square root of 3 over 2,
then add integer multiples of 2 pi.
PTS: 1
DIF: Average
REF: Page 950
OBJ: 13-4.1 Finding Trigonometric Inverses
TOP: 13-4 Inverses of Trigonometric Functions
9. ANS: C
−1
Using the inverse sine function we get θ = sin (0.3) ≈ 17.5°.
Because 90° < θ < 180° we need to find the angle in Quadrant II that has
the same sine value as 17.5°.
θ ≈ 180° − 17.5° = 162.5°
Feedback
A
B
C
D
Check the range given in the problem.
Find the reference angle from the x-axis.
Correct!
Check the range given in the problem.
PTS: 1
DIF: Average
REF: Page 952
OBJ: 13-4.4 Solving Trigonometric Equations
TOP: 13-4 Inverses of Trigonometric Functions
10. ANS: A
[1.] The angle of elevation, θ, may be found using the tangent function, tan θ =
opposite
.
adjacent
The opposite side is the distance from eye level to the top of the pyramid, or 495 feet.
The adjacent side is the distance from the archaeologist to the base of the pyramid, or 260 ft.
tan θ = 495
260
θ = tan−1 495 ≈ 62°
260
[2.] The height of the pyramid, h, from the archaeologist’s eye-level to the top, may be found using
the tangent function, tan θ =
h . The angle of elevation is 50°. The adjacent side is the
adjacent
distance from the archaeologist to the base of the pyramid, or 400 ft.
tan 50° =
h
400
h = 400 tan 50° ≈ 477 ft
Adding 5 ft to this gives a total height of 482 ft.
Feedback
A
B
C
D
Correct!
The tangent of an angle is the opposite side divided by the adjacent side.
The tangent of an angle is the opposite side divided by the adjacent side.
The tangent of an angle is the opposite side divided by the adjacent side.
PTS: 1
11. ANS: A
area = 12 bh
DIF:
h = asin C
1
2
1
2
absinC
area = (6)(8) sin 73 = 23.0
area =
ft
2
Advanced
TOP: 13-4 Inverses of Trigonometric Functions
Write the area formula.
Solve for h using trigonometric ratios.
Substitute.
Feedback
A
B
C
Correct!
The first term of the area formula for a triangle is one half.
The area of a triangle is one half the product of the lengths of two of the sides
and the sine of their included angle.
D
The area of a triangle is one half the product of the lengths of two of the sides
and the sine of their included angle.
PTS: 1
DIF: Basic
REF: Page 958
OBJ: 13-5.1 Determining the Area of a Triangle
12. ANS: B
Step 1 Find the third angle measure.
TOP: 13-5 The Law of Sines
m∠M + m∠N + m∠P = 180°
m∠M + 118° + 33° = 180°
m∠M = 29°
Step 2 Find the unknown side lengths.
sin M = sin N
m
n
sin 29° = sin 118°
15
n
n = 15 sin 118°
sin 29°
n ≈ 27.3
Law of Sines
Substitute.
Solve for the unknown
side.
sin M = sin P
m
p
sin 29° = sin 33°
15
p
p = 15 sin 33°
sin 29°
p ≈ 16.9
Feedback
A
B
C
D
Use the Law of Sines correctly to find the measure of n.
Correct!
Use the Law of Sines correctly.
Use the Law of Sines correctly to find the measure of p.
PTS: 1
DIF: Average
REF: Page 959
OBJ: 13-5.2 Using the Law of Sines for AAS and ASA
TOP: 13-5 The Law of Sines
NUMERIC RESPONSE
13. ANS:
3
5
PTS: 1
14. ANS: 39.69
PTS:
1
DIF:
Average
TOP: 13-1 Right-Angle Trigonometry
DIF:
Advanced
TOP: 13-6 The Law of Cosines
MATCHING
15. ANS: A
PTS: 1
TOP: 13-1 Right-Angle Trigonometry
DIF:
Basic
REF: Page 929
16. ANS:
TOP:
17. ANS:
TOP:
18. ANS:
TOP:
19. ANS:
TOP:
20. ANS:
TOP:
I
PTS: 1
13-1 Right-Angle Trigonometry
H
PTS: 1
13-1 Right-Angle Trigonometry
B
PTS: 1
13-1 Right-Angle Trigonometry
G
PTS: 1
13-1 Right-Angle Trigonometry
E
PTS: 1
13-1 Right-Angle Trigonometry
DIF:
Basic
REF: Page 932
DIF:
Basic
REF: Page 932
DIF:
Basic
REF: Page 929
DIF:
Basic
REF: Page 932
DIF:
Basic
REF: Page 929
21. ANS:
TOP:
22. ANS:
TOP:
23. ANS:
TOP:
24. ANS:
TOP:
25. ANS:
TOP:
26. ANS:
TOP:
H
PTS: 1
13-2 Angles of Rotation
C
PTS: 1
13-2 Angles of Rotation
F
PTS: 1
13-2 Angles of Rotation
A
PTS: 1
13-2 Angles of Rotation
G
PTS: 1
13-2 Angles of Rotation
E
PTS: 1
13-2 Angles of Rotation
DIF:
Basic
REF: Page 937
DIF:
Basic
REF: Page 936
DIF:
Basic
REF: Page 936
DIF:
Basic
REF: Page 936
DIF:
Basic
REF: Page 937
DIF:
Basic
REF: Page 936