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3 APPLY
ASSIGNMENT GUIDE
GUIDED PRACTICE
Vocabulary Check
BASIC
Day 1: pp. 562–565 Exs. 10–38
even
Day 2: pp. 562–566 Exs. 11–39
odd, 47–51, 53, 54, 56–60,
Quiz 2 Exs. 1–7
Concept Check
AVERAGE
Day 1: pp. 562–565 Exs. 10–38
even, 40, 42, 46
Day 2: pp. 562–566 Exs. 11–39
odd, 43, 44, 47–51, 53, 54,
56–60, Quiz 2 Exs. 1–7
ADVANCED
Day 1: pp. 562–565 Exs. 10–38
even, 40–42, 46
Day 2: pp. 562–566 Exs. 11–39
odd, 43, 44, 47–60, Quiz 2
Exs. 1–7
Skill Check
✓
✓
✓
1. The sine of ™A is the ratio
of the length of the leg
opposite ™A to the length
of the hypotenuse. The
cosine of ™A is the ratio
of the length of the leg
adjacent to ™A to the
length of the hypotenuse.
The tangent of ™A is the
ratio of the length of the leg
opposite ™A to the length
of the leg adjacent to ™A.
In Exercises 1 and 2, use the diagram at the right.
B
1. Use the diagram to explain what is meant by the
sine, the cosine, and the tangent of ™A.
A
See margin.
2. ERROR ANALYSIS A student says that
sin D > sin A because the side lengths of
¤DEF are greater than the side lengths of
¤ABC. Explain why the student is incorrect.
See margin.
In Exercises 3–8, use the diagram shown at the
right to find the trigonometric ratio.
3
4. cos A }5} = 0.6
6. sin B }3} = 0.6
5
3
8. tan B }} = 0.75
4
4
3. sin A }} = 0.8
5
4
5. tan A }3} ≈ 1.3333
4
7. cos B }5} = 0.8
9.
F
37°
C
37°
D
E
A
5
3
C
4
ESCALATORS One early escalator built in
1896 rose at an angle of 25°. As shown in the
diagram at the right, the vertical lift was 7 feet.
Estimate the distance d a person traveled on
this escalator. about 17 ft
B
d
7 ft
25ⴗ
BLOCK SCHEDULE
pp. 562–565 Exs. 10–38 even
(with 9.4)
pp. 562–566 Exs. 11–39 odd,
43, 44, 47–51, 53, 54, 56–60,
Quiz 2 Exs. 1–7 (with 9.6)
EXERCISE LEVELS
Level A: Easier
10–27
Level B: More Difficult
28–54
Level C: Most Difficult
55
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 820.
10. sin R ≈ 0.8491; cos R ≈
0.5283; tan R ≈ 1.6071;
sin S ≈ 0.5283; cos S ≈
0.8491; tan S ≈ 0.6222
11. sin A = 0.8; cos A = 0.6;
tan A ≈ 1.3333;
sin B = 0.6; cos B = 0.8;
tan B = 0.75
FINDING TRIGONOMETRIC RATIOS Find the sine, the cosine, and the
tangent of the acute angles of the triangle. Express each value as a
decimal rounded to four places.
10.
11.
R
53
28
10
13.
14.
E
562
24
HOMEWORK HELP
562
2
兹13
Z
Y
3
See margin.
STUDENT HELP
Example 1: Exs. 10–15,
28–36
Example 2: Exs. 10–15,
28–36
Example 3: Exs. 34–36
Example 4: Exs. 34–36
Example 5: Exs. 16–27
Example 6: Exs. 37–42
Example 7: Exs. 37–42
8
C
6
A
7
HOMEWORK CHECK
To quickly check student understanding of key concepts, go
over the following exercises:
Exs. 12, 24, 30, 34, 38, 48. See also
the Daily Homework Quiz:
• Blackline Master (Chapter 9
Resource Book, p. 84)
•
Transparency (p. 68)
12. X
S
45
T
B
G
1
J
15.
L
F
兹5
25
J
D
H
5
3
2
兹34
K
See margin.
See margin.
See margin.
CALCULATOR Use a calculator to approximate the given value to four
decimal places.
16. sin 48° 0.7431
17. cos 13° 0.9744
18. tan 81° 6.3138
19. sin 27° 0.4540
20. cos 70° 0.3420
21. tan 2° 0.0349
22. sin 78° 0.9781
23. cos 36° 0.8090
24. tan 23° 0.4245
25. cos 63° 0.4540
26. sin 56° 0.8290
27. tan 66° 2.2460
Chapter 9 Right Triangles and Trigonometry
USING TRIGONOMETRIC RATIOS Find the value of each variable. Round
decimals to the nearest tenth.
28.
29.
x
30.
34
t
6
4
s
23ⴗ
36ⴗ
s
37ⴗ
r
s ≈ 31.3; t ≈ 13.3
y
r ≈ 4.9; s ≈ 2.9
x ≈ 10.0; y ≈ 8.0
31.
32.
u
65ⴗ
8
t
t ≈ 7.3; u ≈ 3.4
33.
9
w
70ⴗ
x
6
22ⴗ
v
y
v ≈ 9.6; w ≈ 3.3
x ≈ 16.0; y ≈ 14.9
2. The value of a trigonometric ratio
depends only on the measure of
the acute angle, not on the
particular right triangle used to
compute the value.
12. sin X ≈ 0.8321; cos X ≈ 0.5547;
tan X = 1.5; sin Y ≈ 0.5547;
cos Y ≈ 0.8321; tan Y ≈ 0.6667
13. sin D = 0.28; cos D = 0.96;
tan D ≈ 0.2917; sin F = 0.96;
cos F = 0.28; tan F ≈ 3.4286
14. sin G ≈ 0.8944; cos G ≈ 0.4472;
tan G = 2; sin H ≈ 0.4472;
cos H ≈ 0.8944; tan H = 0.5
15. sin J ≈ 0.8575; cos J ≈ 0.5145;
tan J ≈ 1.6667; sin K ≈ 0.5145;
cos K ≈ 0.8575; tan K = 0.6
FINDING AREA Find the area of the triangle. Round decimals to the
nearest tenth.
36. 34.9 m 2
35. 41.6 m 2
34. 4 cm 2
8m
60⬚
45⬚
4 cm
FOCUS ON
APPLICATIONS
37.
38.
RE
FE
L
AL I
30⬚
11 m
12 m
WATER SLIDE The angle of elevation from
the base to the top of a waterslide is about 13°.
The slide extends horizontally about 58.2 meters.
Estimate the height h of the slide. about 13.4 m
SURVEYING To find the distance d
from a house on shore to a house on an
island, a surveyor measures from the
house on shore to point B, as shown in
the diagram. An instrument called a
transit is used to find the measure of
™B. Estimate the distance d. about 36 m
13ⴗ
h
58.2 m
40 m
B
APPLICATION NOTE
EXERCISE 38 Trigonometry is
applied extensively in surveying
applications. For example, the Great
Trigonometric Survey of India was
done in the 19th century to map out
the country and to measure the
height of Mount Everest, the tallest
mountain in the world. In 1852, the
survey measured Mount Everest
to be 29,002 feet, which is very
close to the height of 29,035 feet
measured in 1999.
42ⴗ
d
WATER SLIDE
Even though riders
on a water slide may travel
at only 20 miles per hour, the
curves on the slide can
make riders feel as though
they are traveling much
faster.
39.
SKI SLOPE Suppose you stand at the top of a ski slope and look down
at the bottom. The angle that your line of sight makes with a line drawn
horizontally is called the angle of depression, as shown below. The vertical
drop is the difference in the elevations of the top and the bottom of the slope.
Find the vertical drop x of the slope in the diagram. Then estimate the
distance d a person skiing would travel on this slope. 482 ft; about 1409 ft
elevation
5500 ft
angle of depression
d
elevation
5018 ft
20ⴗ
x vertical
drop
9.5 Trigonometric Ratios
563
563
FOCUS ON
APPLICATIONS
40.
COMMON ERROR
EXERCISE 40 Students may
have difficulty applying the correct
ratio when solving this problem.
Remind them that the sine or cosine
ratios cannot be used unless the
hypotenuse of the triangle is known.
BC
a
b
}
}
52. Given: sin A = , cos A = }},
c
c
a
tan A = }}
b
sin A
Prove: tan A = }}
cos A
Statements (Reasons)
a
a
b
1. sin A = }}, cos A = }}, tan A = }}
b
c
c
(Given)
sin A a b
2. }} = }} ÷ }} (Substitution
co s A c c
prop. of equality)
a b a
3. }} ÷ }} = }} (Dividing and
c c b
simplifying fractions)
sin A
4. tan A = }} (Substitution
cos A
prop. of equality)
RE
L
AL I
FE
46. måB = 95°, so †ABC is not a right
triangle, as required by the definition of the tangent ratio. To find
BC, draw an altitude, B
苶D
苶, from B
to A
苶C苶. †ABD is a 30°-60°-90°
triangle with hypotenuse 18, so
the shorter leg, B
苶D
苶, has length 9.
Then use right †BCD and the
BD
equation sin 55° = }} to find BC.
LUNAR CRATERS
Because the moon
has no atmosphere to
protect it from being hit by
meteorites, its surface is
pitted with craters. There is
no wind, so a crater can
remain undisturbed for
millions of years––unless
another meteorite crashes
into it.
a
b
43. sin A = }} ; cos A = }};
c
c
a
b
tan A = }}; sin B = }};
b
c
b
a
cos B = }} ; tan B = }}
a
c
55ⴗ
d
55ⴗ
500 m
42.
LUGGAGE DESIGN Some luggage
pieces have wheels and a handle so that
the luggage can be pulled along the ground.
Suppose a person’s hand is about 30 inches
from the floor. About how long should the
handle be on the suitcase shown so that
it can roll at a comfortable angle of 45°
with the floor? about 16.4 in.
x
26 in.
30 in.
45ⴗ
BUYING AN AWNING Your family room
has a sliding-glass door with a southern exposure.
You want to buy an awning for the door that will
be just long enough to keep the sun out when it
is at its highest point in the sky. The angle of
elevation of the sun at this point is 70°, and the
height of the door is 8 feet. About how far should
the overhang extend? about 2.9 ft
sun’s ray
x
8 ft
70⬚
44. The tangent of one acute
angle of a right triangle is
the reciprocal of the
tangent of the other acute CRITICAL THINKING In Exercises 43 and 44, use the diagram.
angle. The sine of one
43. Write expressions for the sine, the cosine, and the tangent
acute angle of a right
of each acute angle in the triangle.
triangle is equal to the
cosine of the other acute 44. Writing Use your results from Exercise 43 to explain
angle, and the cosine of
c
how the tangent of one acute angle of a right triangle
one acute angle is equal
is related to the tangent of the other acute angle. How
to the sine of the other
are the sine and the cosine of one acute angle of a right
acute angle.
45. Procedures may vary. One
method is to reason that
since the tangent ratio is
45.
equal to the ratio of the
lengths of the legs, the
tangent is equal to 1 when
the legs are equal in
length, that is, when the
triangle is a 45°–45°–90°
triangle. Tan A > 1 when
m™A > 45°, and tan A < 1
when m™A < 45°, since 46.
increasing the measure of
™A increases the length of
the opposite leg and
decreasing the measure of
™A decreases the length
of the opposite leg.
564
564
sun’s ray
about 714 m
41.
EXERCISE 46 Students may
want to apply the right triangle
ratios to any triangle or to the right
angle of the triangle. Remind students to check that a triangle is
right before using the ratios and
that the ratios apply only to the
acute angles.
SCIENCE CONNECTION Scientists can
measure the depths of craters on the
moon by looking at photos of shadows.
The length of the shadow cast by the
edge of a crater is about 500 meters.
The sun’s angle of elevation is 55°.
Estimate the depth d of the crater.
triangle related to the sine and the cosine of the other
acute angle?
A
b
B
a
C
TECHNOLOGY Use geometry software to construct a right triangle.
Use your triangle to explore and answer the questions below. Explain
your procedure.
•
•
•
For what angle measure is the tangent of an acute angle equal to 1?
For what angle measures is the tangent of an acute angle greater than 1?
For what angle measures is the tangent of an acute angle less than 1?
Æ
ERROR ANALYSIS To find the length of BC
B
in the diagram at the right, a student writes
18
18
tan 55° = ᎏᎏ. What mistake is the student
BC
making? Show how the student can find BC.
(Hint: Begin by drawing an altitude from
Æ
B to AC.) See margin.
Chapter 9 Right Triangles and Trigonometry
30°
A
55°
C
B
PROOF Use the diagram of ¤ABC. Complete
47.
the proof of the trigonometric identity below.
c
(sin A)2 + (cos A)2 = 1
a
c
A
Q⬘
C
b
b
c
56.
a
PROVE 䉴 (sin A)2 + (cos A)2 = 1
Q
3
2
2
48. (sin 30°) + (cos 30°) =
冉冊 冉 冊
1 2
}} +
2
1
3
}} + }} =
4
4
Statements
a
b
1. sin A = ᎏᎏ, cos A = ᎏᎏ
c
c
2. a2 + b2 = c 2
a2
b2
3. ᎏᎏ2 + ᎏᎏ2 = 1
c
c
兹3苶 2
} =
2
1✓
2
2
49. (sin 45°) + (cos 45°) =
冉 冊 冉 冊
兹2苶 2
兹2苶 2
} + } =
2
2
2
2
}} + }} = 1 ✓
4
4
4.
冉 冊 冉冊
2
2
5. (sin A)2 + (cos A)2 = 1
50. (sin 60°) 2 + (cos 60°) 2 =
兹3苶 2
1 2
} + }} =
2
2
3
1
}} + }} = 1 ✓
4
4
冉ᎏacᎏ冊 + 冉ᎏbcᎏ冊 = 1
10
6
GIVEN 䉴 sin A = ᎏᎏ, cos A = ᎏᎏ
P⬘
5
P 4
Reasons
?㛭㛭㛭
1. 㛭㛭㛭㛭㛭
Given
?㛭㛭㛭
2. 㛭㛭㛭㛭㛭
Pythagorean Thm.
?㛭㛭㛭
3. 㛭㛭㛭㛭㛭
Division prop. of equality
8
R⬘
R
4. A property of exponents
?㛭㛭㛭
5. 㛭㛭㛭㛭㛭
Substitution prop. of equality
DEMONSTRATING A FORMULA Show that (sin A)2 + (cos A)2 = 1 for the
given angle measure.
48. m™A = 30°
49. m™A = 45°
50. m™A = 60°
51. m™A = 13°
51. (sin 13°) 2 + (cos 13°) 2 ≈
2
2
(0.2250) + (0.9744) ≈ 1 ✓ 52.
PROOF Use the diagram in Exercise 47. Write a two-column proof of the
sin A
following trigonometric identity: tan A = ᎏᎏ. See margin.
cos A
Test
Preparation
53. MULTIPLE CHOICE Use the diagram at the right.
Find CD. D
A
¡
D
¡
8 cos 25°
8
ᎏᎏ
sin 25°
B
¡
E
¡
8 sin 25°
C
¡
D
8
8 tan 25°
25ⴗ
C
8
ᎏᎏ
cos 25°
54. MULTIPLE CHOICE Use the diagram at the
right. Which expression is not equivalent to AC? C
A
¡
D
¡
★ Challenge
EXTRA CHALLENGE
www.mcdougallittell.com
55.
BC sin 70°
BA
ᎏᎏ
tan 20°
B
¡
E
¡
BC cos 20°
C
¡
BC
ᎏᎏ
tan 20°
E
B
A
160ⴗ
C
BA tan 70°
PARADE You are at a parade
looking up at a large balloon floating
directly above the street. You are
60 feet from a point on the street
directly beneath the balloon. To see the
top of the balloon, you look up at an
angle of 53°. To see the bottom of the
balloon, you look up at an angle of 29°.
Estimate the height h of the balloon
to the nearest foot. 46 ft
h
ADDITIONAL PRACTICE
AND RETEACHING
For Lesson 9.5:
53ⴗ
29ⴗ
60 ft
9.5 Trigonometric Ratios
565
• Practice Levels A, B, and C
(Chapter 9 Resource Book, p. 72)
• Reteaching with Practice
(Chapter 9 Resource Book, p. 75)
•
See Lesson 9.5 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
565
4 ASSESS
DAILY HOMEWORK QUIZ
MIXED REVIEW
56. An enlargement; 2;
Q§R§ = 10 and
P§R§ = 8
Transparency Available
Find the sine, the cosine, and the
tangent of the acute angles of the
triangle. Express each value as a
decimal rounded to four places.
A
1.
56. SKETCHING A DILATION ¤PQR is mapped onto ¤P§Q§R§ by a dilation.
In ¤PQR, PQ = 3, QR = 5, and PR = 4. In ¤P§Q§R§, P§Q§ = 6. Sketch the
dilation, identify it as a reduction or an enlargement, and find the scale factor.
Then find the length of Q§R§ and P§R§. (Review 8.7)
57. FINDING LENGTHS Write similarity
N
statements for the three similar triangles in
the diagram. Then find QP and NP. Round
decimals to the nearest tenth. (Review 9.1)
7
¤MNP ~ ¤MQN ~ ¤NQP; QP ≈ 3.3; NP ≈ 7.8 M
q
15
P
35
12
B
37
PYTHAGOREAN THEOREM Find the unknown side length. Simplify answers
that are radicals. Tell whether the side lengths form a Pythagorean triple.
(Review 9.2 for 9.6)
C
sin B ≈ 0.9459, cos B ≈ 0.3243,
tan B ≈ 2.9167, sin C ≈ 0.3243,
cos C ≈ 0.9459, tan C ≈ 0.3429
58.
59.
x
Use the diagram. Round decimals
to the nearest tenth.
60.
65
x
42.9
95
x
50
5兹69
苶; no
168; yes 193
70
82.1; no
y
x
32°
QUIZ 2
8m
Sketch the figure that is described. Then find the requested information.
Round decimals to the nearest tenth. (Lesson 9.4)
2. Find the value of each variable.
x ≈ 5 m, y ≈ 9.4 m
1. The side length of an equilateral triangle is 4 meters. Find the length of
an altitude of the triangle. 3.5 m
3. Find the area of the triangle.
20 m
Self-Test for Lessons 9.4 and 9.5
2
2. The perimeter of a square is 16 inches. Find the length of a diagonal. 5.7 in.
3. The side length of an equilateral triangle is 3 inches. Find the area of
the triangle. 3.9 in.2
EXTRA CHALLENGE NOTE
Challenge problems for
Lesson 9.5 are available in
blackline format in the Chapter 9
Resource Book, p. 80 and at
www.mcdougallittell.com.
Find the value of each variable. Round decimals to the nearest tenth.
(Lesson 9.5)
4.
x
y
x ≈ 15.6; y ≈ 11.9
7.
tangent of the angle of elevation
and multiply it by your distance
from the tree.
566
10
40ⴗ
ADDITIONAL TEST
PREPARATION
1. WRITING Describe how to find
the height of a tree if you know
the angle of elevation and your
distance from the tree. Find the
ADDITIONAL RESOURCES
An alternative Quiz for Lessons
9.4–9.5 is available in the Chapter 9
Resource Book, p. 81.
5.
566
y
y
x
25ⴗ
20
x
62ⴗ
18
x ≈ 9.3; y ≈ 22.1
x ≈ 8.5; y ≈ 15.9
HOT-AIR BALLOON The ground
crew for a hot-air balloon can see the
balloon in the sky at an angle of
elevation of 11°. The pilot radios to the
crew that the hot-air balloon is 950 feet
above the ground. Estimate the horizontal
distance d of the hot-air balloon from the
ground crew. (Lesson 9.5) about 4887 ft
Chapter 9 Right Triangles and Trigonometry
6.
Not drawn to scale
ground
crew
11ⴗ
d
950 f t