Download Pre-Test

Pre-Test
Name _________________________________________________________
Date _________________________
1. Write the trigonometric ratios for ⬔A. Write your answers as simplified fractions.
A
10 cm
6 cm
C
8 cm
B
sin A ⴝ
8
4
ⴝ
10 5
cos A ⴝ
6
3
ⴝ
10 5
tan A ⴝ
8 4
ⴝ
6 3
2. Write the trigonometric ratios for ⬔B. Write your answers as simplified fractions.
A
10 cm
6 cm
C
8 cm
B
sin B ⴝ
6
3
ⴝ
10 5
cos B ⴝ
8
4
ⴝ
10 5
tan B ⴝ
6 3
ⴝ
8 4
4
Use trigonometric ratios to find the value of x. Show all your work and round your answer to
the nearest tenth.
3.
4.
© 2009 Carnegie Learning, Inc.
x
20 mm
48⬚
5.
12 in.
39⬚
x
64⬚
x
15 ft
x
sin 48º ⴝ
20
cos 39º ⴝ
20(sin 48º) ⴝ x
xⴝ
14.9 mm ⬇ x
12
x
12
cos 39 º
tan 64º ⴝ
x
15
15(tan 64º) ⴝ x
30.8 ft ⬇ x
x ⬇ 15.4 in.
Chapter 4 ● Assessments
83
Pre-Test
PAGE 2
6. The cosine of ⬔A is approximately 0.62. Estimate the measure of ⬔A.
The value of the cosine of an acute angle decreases as the measure of the angle increases.
Because cos 45º ⬇ 0.71 and cos 60º = 0.5, m⬔A should be between 45º and 60º.
7. Josh is using a clinometer to determine the height of a building. He places the clinometer 50 feet from the
base of the building and measures the angle of elevation to be 72º. Draw a diagram that models this situation.
Then, find the height of the building. Round your answer to the nearest foot.
x
tan 72º ⴝ
50
50(tan 72º ) ⴝ x
x
72⬚
154 ⬇ x
clinometer
50 ft
The height of the building is about 154 feet.
8. Louise is using a clinometer to determine the width of a gorge. She stands on one side of the gorge,
and uses a clinometer to measure the angle of depression to a point directly across the gorge on the
other side. Louise is at an elevation of 3458 feet and the point on the other side of the gorge is at an
elevation of 2129 feet. She measures a 39º angle of depression. Draw a diagram that models this
situation. Then, find the width of the gorge. Round your answer to the nearest foot.
4
tan 51º ⴝ
x
1329
1329(tan 51º ) ⴝ x
1641⬇ x
39⬚
3458 feet 51⬚
− 2129 feet
1329 feet
x
© 2009 Carnegie Learning, Inc.
The width of the gorge is about 1641 feet.
Louise
84
Chapter 4 ● Assessments
Post-Test
Name _________________________________________________________
Date _________________________
1. Write the trigonometric ratios for ⬔A. Write your answers as simplified fractions.
A
sin A ⴝ
36 12
ⴝ
39 13
cos A ⴝ
15
5
ⴝ
39 13
tan A ⴝ
36 12
ⴝ
15
5
39 mm
15 mm
C
B
36 mm
2. Write the trigonometric ratios for ⬔B. Write your answers as simplified fractions.
A
39 mm
15 mm
C
B
36 mm
sin B ⴝ
15
5
ⴝ
39 13
cos B ⴝ
36 12
ⴝ
39 13
tan B ⴝ
15
5
ⴝ
36 12
4
© 2009 Carnegie Learning, Inc.
Use trigonometric ratios to find the value of x. Show all your work and round your answer
to the nearest tenth.
3.
4.
x
8 mm
26⬚
40 in.
58⬚
x
5.
x
71⬚
27 ft
x
8
8(sin 26º) ⴝ x
sin 26º ⴝ
3.5 mm ⬇ x
40
cos 58º ⴝ
x
40
xⴝ
cos 58 º
tan 71º ⴝ
x
27
27(tan 71º) ⴝ x
78.4 ft ⬇ x
x ⬇ 75.5 in.
Chapter 4 ● Assessments
85
Post-Test
PAGE 2
6. The sine of ⬔A is approximately 0.63. Estimate the measure of ⬔A.
The value of the sine of an acute angle increases as the measure of the angle increases. Because
sin 30º = 0.5 and sin 45º ⬇ 0.71, m⬔A should be between 30º and 45º.
7. Pierre is using a clinometer to determine the height of the Eiffel Tower in Paris. He places the clinometer
80.8 meters from the base of the tower and measures the angle of elevation to be 76º. Draw a diagram
that models this situation. Then, find the height of the Eiffel Tower. Round your answer to the nearest
meter.
x
tan 76º ⴝ
80.8
80.8(tan 76º) ⴝ x
324 ⬇ x
x
The height of the Eiffel Tower is about 324 meters.
76⬚
clinometer
8. Dominique is using a clinometer to determine the horizontal distance between her office and her friend
Chantel’s office. She stands at her office window, and uses a clinometer to measure the angle of
depression to Chantel’s office window. Dominique is at an elevation of 80 feet and Chantel’s office is at
an elevation of 36 feet. She measures a 25º angle of depression. Draw a diagram that models this
situation. Then, find the horizontal distance between the offices. Round your answer to the nearest foot.
44
tan 25º ⴝ
x
Dominique’s
office
x
⬚
25
44
ⴝx
tan 25º
94 ⬇ x
The horizontal distance between Chantel’s office and Dominque’s office is about 94 feet.
86
Chapter 4 ● Assessments
80 feet
− 36 feet
44 feet
Chantel’s
office
© 2009 Carnegie Learning, Inc.
4
80.8 m
Mid-Chapter Test
Name _________________________________________________________
Date _________________________
Use trigonometric ratios to find the value of x. Show all your work and round your answer to
the nearest tenth.
1.
2.
x
28⬚
13 mm
x
57⬚
5 ft
tan 28º ⴝ
x
13
13(tan 28º) ⴝ x
6.9 mm ⬇ x
cos 57 º ⴝ
5
x
5
ⴝx
cos 57º
9.2 ft ⬇ x
3.
x
14⬚
48 yd
4.
27 cm
x
4
35⬚
© 2009 Carnegie Learning, Inc.
cos 14º ⴝ
x
48
48(cos 14º ) ⴝ x
46.6 yd ⬇ x
tan 35º ⴝ
xⴝ
27
x
27
tan 35 º
x ⬇ 38.6 cm
Chapter 4 ● Assessments
87
Mid-Chapter Test
PAGE 2
5. DeJuan’s house is 18 miles due south of Jamie’s house. Leslie’s house is due east of DeJuan’s house
and southeast of Jamie’s house. Use the following figure to determine how far is Leslie’s house from
DeJuan’s house? Round your answer to the nearest tenth of a mile if necessary.
Jamie’s
house
N
W
E
S
18 mi
65⬚
DeJuan’s
house
tan 65º ⴝ
xⴝ
Leslie’s
house
18
x
18
tan 65º
x ⬇ 8.4
Leslie’s house is about 8.4 miles from DeJuan’s house.
6. Michelle’s house is 22 miles due north of Jackie’s house and northeast of Patrick’s house. Patrick’s
house is due west of Jackie’s house. Use the following figure to determine how far is Michelle’s house
from Patrick’s house? Round your answer to the nearest tenth of a mile if necessary.
Michelle’s
house
N
W
⬚
38
E
S
22 mi
Patrick’s
house
cos 38º ⴝ
xⴝ
Jackie’s
house
22
x
22
cos 38º
x ⬇ 27.9
Michelle’s house is about 27.9 miles from Patrick’s house.
88
Chapter 4 ● Assessments
© 2009 Carnegie Learning, Inc.
4
Mid-Chapter Test
PAGE 3
Name _________________________________________________________
Date _________________________
7. Your teacher asks you to find the area of a regular octagon. The length of each side of the octagon is
6 millimeters. You know that you need to find the length of the apothem to calculate the area. Use a
trigonometric ratio to find the length of the apothem and then find the area of the regular octagon.
Round your answer to the nearest tenth if necessary. Hint: The area of a regular polygon can be found
1
by using the formula A ⫽ aP .
2
(
)
22.5⬚
a
6 mm
Length of apothem: tan 22.5º ⴝ
aⴝ
3
a
3
tan 22.5º
a ⬇ 7.2 mm
Area of octagon:
1
A ⴝ (7.2)(48)
2
4
A ⴝ 172.8 mm2
© 2009 Carnegie Learning, Inc.
The length of the apothem is about 7.2 millimeters and the area of the octagon is about
172.8 square millimeters.
8. The cosine of ⬔A is approximately 0.79. Estimate the measure of ⬔A.
The value of the cosine of an acute angle decreases as the measure of the angle increases.
Because cos 30º ⬇ 0.87 and cos 45º ⬇ 0.71, m⬔A should be between 30º and 45º.
Chapter 4 ● Assessments
89
© 2009 Carnegie Learning, Inc.
4
90
Chapter 4 ● Assessments
End of Chapter Test
Name _________________________________________________________
Date _________________________
Use trigonometric ratios to find the value of x. Show all your work and round your answer to
the nearest tenth.
1.
2.
55⬚
x
8 in.
73⬚
27 ft
sin 55º ⴝ
x
27
x
tan 73º ⴝ
xⴝ
27
xⴝ
sin 55º
x ⬇ 33.0 ft
8
x
8
tan 73º
x ⬇ 2.4 in.
3.
4.
x
x
61⬚
48⬚
15 mm
© 2009 Carnegie Learning, Inc.
sin 48º ⴝ
x
15
15(sin 48º ) ⴝ x
11.1 mm ⬇ x
4
54 mm
cos 61º ⴝ
x
54
54(cos 61º ) ⴝ x
26.2 mm ⬇ x
Chapter 4 ● Assessments
91
End of Chapter Test
PAGE 2
5. A ramp at a skateboard park is 30 feet long and has a 30º incline. How high is the top of the ramp?
x
sin 30º ⴝ 30
30 ft
x
30⬚
30(sin 30º) ⴝ x
15 ⴝ x
The top of the ramp is 15 feet high.
6. Use a trigonometric ratio to find the width of the following rectangle. Round your answer to the nearest tenth of
a centimeter.
x
59⬚
37 cm
x
cos 59º ⴝ 37
37(cos 59º) ⴝ x
19.1 ⬇ x
The width of the rectangle is about 19.1 centimeters.
7. Paul is making a mosaic design on the floor in a museum. He needs the triangle in the design to cover
4.25 square feet. Does the triangle he has sketched in his design meet the specifications?
2 ft
⬚
26
Base of the triangle:
tan 26º ⴝ
2
x
2
ⴝx
tan 26º
Area of the triangle:
Aⴝ
1
(4.10)(2)
2
ⴝ 4.10
4.10 ⬇ x
The area of the triangle is about 4.10 square feet. So, it will not cover an area of
4.25 square feet.
92
Chapter 4 ● Assessments
© 2009 Carnegie Learning, Inc.
4
End of Chapter Test
PAGE 3
Name _________________________________________________________
Date _________________________
8. A regular pentagon is inscribed in a circle with a radius of 10 inches. What is the length of each side of
the pentagon? Round your answer to the nearest tenth of an inch.
x
sin 36º ⴝ 10
10(sin 36º) ⴝ x
5.9 ⬇ x
36⬚
The length of one side of the regular pentagon is about 11.8 inches.
9. Olivia uses a clinometer to measure the height of the observation deck of the Seattle Space Needle.
She stands on the observation deck, and uses the clinometer to measure the angle of depression to
the top of a building that is 100 feet tall. The building is 466.5 feet from the Space Needle. Olivia
measures a 42º angle of depression. Draw a diagram that models this situation. Then, find the height
of the observation deck. Round your answer to the nearest foot.
4
Height of the observation deck ⴝ x + 100 feet
tan 48º ⴝ
© 2009 Carnegie Learning, Inc.
xⴝ
42⬚
48⬚
466.5
x
x
466.5
tan 48º
x ⬇ 420
The height of the observation deck is about 520 feet.
100 ft
466.5 ft
Chapter 4 ● Assessments
93
End of Chapter Test
PAGE 4
10. Bernard lives in an apartment building in the city, and he can see a skyscraper from his living room
window. He would like to know how far his apartment building is from the skyscraper. He uses a
clinometer to measure the angle of elevation from his apartment to the top of the skyscraper. The
angle of elevation is 38º. He knows that the skyscraper is 630 feet tall and the height of his living room
window is 200 feet. Draw a diagram that models this situation. Then, find the distance between
Bernard’s apartment and the skyscraper. Round your answer to the nearest foot.
630 ft
38⬚
x
200 ft
tan 38º ⴝ
xⴝ
4
430
x
430
tan 38º
x ⬇ 550
© 2009 Carnegie Learning, Inc.
The distance from Bernard’s living room to the skyscraper is about 550 feet.
94
Chapter 4 ● Assessments
Standardized Test Practice
Name _________________________________________________________
Date _________________________
1. Eric is flying an airplane at an altitude of 2200 feet. He sees his house on the ground at a 45º angle of
depression.
45⬚
2200 ft
What is Eric’s horizontal distance from his house at this point?
a. 110 feet
b. 220 feet
c. 1100 feet
d. 2200 feet
2. In the following figure, if tan x ⫽
4
8
, what are sin x and cos x?
15
© 2009 Carnegie Learning, Inc.
x
a. sin x =
17
17
and cos x =
8
15
b. sin x =
8
15
and cos x =
17
17
c. sin x =
15
8
and cos x =
17
17
d. sin x =
8
17
and cos x =
15
8
Chapter 4 ● Assessments
95
Standardized Test Practice
PAGE 2
3. Luis is standing on a street in New York City looking at the top of the Empire State Building with a
30º angle of elevation. He is 767.6 meters from the Empire State Building.
30⬚
767.6 m
How tall is the Empire State Building?
a. 383.8 meters
b. 443.2 meters
c. 664.8 meters
d. 1329.5 meters
4. In the following figure, cos P ⫽ 0.60.
N
P
24 cm
M
What is the length of PN?
a. 0.25 centimeters
b. 14.4 centimeters
c. 40 centimeters
d. 44 centimeters
96
Chapter 4 ● Assessments
© 2009 Carnegie Learning, Inc.
4
Standardized Test Practice
PAGE 3
Name _________________________________________________________
Date _________________________
5. In the following diagram, m⬔B ⫽ 42º and AB ⫽ 25 feet. Which equation can be used to find the value
of x?
A
25 ft
x
42⬚
C
B
a. x = 25(sin 42º)
b. x = 25(cos 42º)
c. x = 25(tan 42º)
d. x =
sin 42º
25
4
6. In the following diagram, a 12-foot slide is attached to a swing set. The slide makes a 65º angle with
the swing set. Which answer most closely represents the height of the top of the slide?
12 ft
sin 65º ⬇ 0.91
65⬚
?
cos 65º ⬇ 0.42
© 2009 Carnegie Learning, Inc.
tan 65º ⬇ 2.14
a. 5.0 feet
b. 5.6 feet
c. 10.9 feet
d. 25.7 feet
Chapter 4 ● Assessments
97
Standardized Test Practice
PAGE 4
7. Which of the following statements is true?
a. As the measure of the angle increases, the value of the sine and the cosine increases.
b. As the measure of the angle increases, the value of the sine and the cosine decreases.
c. As the measure of the angle increases, the value of the sine and the tangent increases.
d. As the measure of the angle increases, the value of the cosine and the tangent decreases.
8. A regular hexagon is inscribed in a circle with a radius of 10 meters. What is the length of each side of
the hexagon?
10 m
a. 5 meters
b. 5冪3 meters
4
c. 10 meters
© 2009 Carnegie Learning, Inc.
d. Cannot be determined
98
Chapter 4 ● Assessments
Standardized Test Practice
PAGE 5
Name _________________________________________________________
Date _________________________
9. The cos A ⬇ 0.67. Which of the following statements must be true?
a. The measure of ⬔A is between 30º and 45º.
b. The measure of ⬔A is between 45º and 60º.
c. The measure of ⬔A is between 60º and 75º.
d. The measure of ⬔A is between 75º and 90º.
10. Which of the following statements is NOT true?
a. The cosine of an acute angle is always less than or equal to one.
b. The sine of an acute angle is always less than or equal to one.
c. The tangent of an acute angle is always less than or equal to one.
d. The value of the sine of an angle divided by the value of the cosine of the angle is equal to the value
of the tangent of the angle.
© 2009 Carnegie Learning, Inc.
4
Chapter 4 ● Assessments
99
© 2009 Carnegie Learning, Inc.
4
100
Chapter 4 ● Assessments