Download 10.6 Solving Right Triangles

Page 1 of 7
10.6
Goal
Solving Right Triangles
Key Words
To solve a right triangle means to find the measures of both acute
angles and the lengths of all three sides. Suppose you know the
lengths of the legs of a right triangle. How would you find the
B
measures of the angles?
• tangent p. 557
In the triangle at the right, the legs have lengths
• sine p. 563
7 and 10, so the tangent of aA is , or 0.7.
Solve a right triangle.
• cosine p. 563
• solve a right triangle
• inverse tangent
7
7
10
A
10
C
You can use the table of trigonometric ratios on page 705 to find
the measure of aA. Or you can use the inverse tangent function
(tan1 x) of a scientific calculator to find the angle measure.
• inverse sine
Look for 0.7 in the
tangent column.
• inverse cosine
Table of Trigonometric Ratios
Sine
38
33
34
Cosine
.8387
.8290
.5446
.5592
Tangent
ON/AC
.6494
.6745
.5878
.6018
.6157
.8090
.7986
.7880
READING TIP
The phrase “tan–1 z” is
read as “the inverse
tangent of z.”
EE
COS
TAN
LOG
LN
.7265
.7536
.7813
1
x
%
DRG
x2
SIN
COS
TAN
yx
INV
EE
LOG
LN
÷
CE/C
π
K
(
)
STO
7
8
9
–
RCL
4
5
6
+
SUM
1
2
3
=
EXC
0
.
+–
On this calculator, you press INV
then TAN to get the inverse tangent.
The angle with a tangent of 0.7
has a measure of about 35°.
Student Help
SIN
INV
35 .5736 .8192 .7002
36
37
38
x
x2
+
Angle
INVERSE TANGENT
B
For any acute angle A of a right triangle:
z
If tan A z, then tan1 z maA.
EXAMPLE
1
A
1
C
Use Inverse Tangent
Use a calculator to approximate the measure
of aA to the nearest tenth of a degree.
B
8
Solution
C
8
10
Since tan A 0.8, tan1 0.8 ma A.
Expression
tan
1
0.8
Calculator keystrokes
0.8
INV
TAN
10
A
Display
or
38.65980825
0.8
ANSWER
Because tan1 0.8 ≈ 38.7, ma A ≈ 38.7.
10.6
Solving Right Triangles
569
Page 2 of 7
IStudent Help
EXAMPLE
2
Solve a Right Triangle
ICLASSZONE.COM
MORE EXAMPLES
More examples at
classzone.com
Find each measure to the nearest tenth.
a. c
b. maB
C
c. ma A
3
2
Solution
(hypotenuse)2 (leg)2 (leg)2
2
2
c
B
a. Use the Pythagorean Theorem to find c.
A
Pythagorean Theorem
2
c 3 2
Substitute.
c 2 13
Simplify.
c 1
3
Find the positive square root.
c ≈ 3.6
Use a calculator to approximate.
b. Use a calculator to find maB.
2
3
Since tan B ≈ 0.6667, maB ≈ tan1 0.6667 ≈ 33.7.
c. aA and aB are complementary, so maA ≈ 90 33.7 56.3.
Use Inverse Tangent
aA is an acute angle. Use a calculator to approximate the
measure of aA to the nearest tenth of a degree.
1. tan A 3.5
2. tan A 2
3. tan A 0.4402
Find the measure of aA to the nearest tenth of a degree.
4.
A
9
B
5. B
6. C
5
17
10
C
C
20
16
B
A
A
Inverse Sine and Inverse Cosine A scientific calculator has
inverse sine (sin–1 x) and inverse cosine (cos–1 x) functions. Use
these inverse functions if you are given the lengths of one leg and
the hypotenuse.
On this calculator, you press
INV
SIN and INV
COS
to get the inverse functions.
570
Chapter 10
Right Triangles and Trigonometry
ON/AC
x
2
INV
SIN
EE
COS
LOG
TAN
LN
x
1
x
%
DRG
CE/C
x2
SIN
COS
TAN
yx
INV
EE
LOG
LN
÷
π
K
(
)
+
STO
7
8
9
–
RCL
4
5
6
+
SUM
1
2
3
=
EXC
0
.
+–
Page 3 of 7
INVERSE SINE AND INVERSE COSINE
For any acute angle A of a right triangle:
B
If sin A y, then sin1 y maA.
If cos A x, then cos1 x maA.
Student Help
STUDY TIP
To use the table of
ratios on p. 705 to
approximate sin1 0.55,
find the number closest
to 0.55 in the sine
column, then read the
angle measure at its left.
EXAMPLE
A
1
y
x
C
Find the Measures of Acute Angles
3
aA is an acute angle. Use a calculator to approximate the
measure of aA to the nearest tenth of a degree.
a. sin A 0.55
b. cos A 0.48
Solution
a. Since sin A 0.55, maA sin1 0.55.
sin1 0.55 ≈ 33.36701297, so maA ≈ 33.4.
b. Since cos A 0.48, maA cos1 0.48.
cos1 0.48 ≈ 61.31459799, so maA ≈ 61.3.
Student Help
VISUAL STRATEGY
H
39.8
19.2
25
50.2
J
EXAMPLE
Solve a Right Triangle
Solve TGHJ by finding each measure. Round
decimals to the nearest tenth.
16
a. maG
G
Solution
The triangle in Example
4 can be labeled in
color, as suggested on
p. 536.
4
b. maH
H
g
25
c. g
J
16
G
16
25
a. Since cos G 0.64, maG cos1 0.64.
cos1 0.64 ≈ 50.2081805, so maG ≈ 50.2.
b. aG and aH are complementary.
maH 90 maG ≈ 90 50.2 39.8
c. Use the Pythagorean Theorem to find g.
(leg)2 (leg)2 (hypotenuse)2
162 g2 252
Pythagorean Theorem
Substitute.
2
Simplify.
2
Subtract 256 from each side.
g 3
6
9
Find the positive square root.
g ≈ 19.2
Use a calculator to approximate.
256 g 625
g 369
10.6
Solving Right Triangles
571
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Use Inverse Sine and Inverse Cosine
aA is an acute angle. Use a calculator to approximate the
measure of aA to the nearest tenth of a degree.
7. sin A 0.5
8. cos A 0.92
10. cos A 0.5
11. sin A 0.25
9. sin A 0.1149
12. cos A 0.45
Solve the right triangle. Round decimals to the nearest tenth.
13. B
3
C
14.
x
6
4
A
E
4
D
15. J
y
z
F
5
G
7
H
10.6 Exercises
Guided Practice
Vocabulary Check
Skill Check
1. Explain what is meant by solving a right triangle.
Tell whether the statement is true or false.
2. You can solve a right triangle given only the lengths of two sides.
3. You can solve a right triangle given only the measure of one
acute angle.
Find the value of x. Round your answer to the nearest tenth.
4.
5.
x
19
4
x
6.
5
13
x
9
Calculator aA is an acute angle. Use a calculator to approximate the
measure of aA to the nearest tenth of a degree.
7. tan A 5.4472
8. sin A 0.8988
9. cos A 0.3846
Solve the right triangle. Round your answers to the nearest tenth.
10. X
y
Z
11.
E
14
4
x
Y
572
Chapter 10
d
60
Right Triangles and Trigonometry
D
12
F
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Practice and Applications
Extra Practice
See p. 694.
Calculator aA is an acute angle. Use a calculator to approximate the
measure of aA to the nearest tenth of a degree.
12. tan A 0.5
13. tan A 1.0
14. tan A 2.5
15. tan A 0.2311
16. tan A 1.509
17. tan A 4.125
P
Solving a Triangle Tell what method you would
use to solve for the indicated measure. Then find
the measure to the nearest tenth.
18. QS
19. maQ
48
20. maS
T
55
S
Inverse Tangent Use the Pythagorean Theorem to find the length
of the hypotenuse. Then use the inverse tangent to find the measure
of aA to the nearest tenth of a degree.
21.
22. B
C
7
2
7
C
A
x
23.
B
x
6
20
A
x
B
C
21
A
Calculator aA is an acute angle. Use a calculator to approximate the
measure of aA to the nearest tenth of a degree.
24. sin A 0.75
25. cos A 0.1518
26. sin A 0.6
27. cos A 0.45
28. cos A 0.1123
29. sin A 0.6364
Ramps In Exercises 30–32, use the information about ramps.
The Uniform Federal Accessibility Standards require that the measure
of the angle used in a wheelchair ramp be less than or equal to 4.76.
length of ramp
ramp angle
horizontal distance
vertical
rise
30. A ramp has a length of 20 feet and a vertical rise of 2.5 feet. Find
the ramp’s horizontal distance and the measure of its ramp angle.
Does this ramp meet the standards?
Homework Help
Example 1:
Example 2:
Example 3:
Example 4:
Exs. 12–17
Exs. 18–23
Exs. 24–29
Exs. 34–39
31. Suppose a ramp has a vertical rise of 4 feet. Give an example of a
possible length of the ramp that meets the standards.
32. Measurement Measure the horizontal distance and the vertical
rise of a ramp near your home or school. Find the measure of the
ramp angle. Does the ramp meet the standards? Explain.
10.6
Solving Right Triangles
573
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33. Space Shuttle The glide angle of a space shuttle is the angle
indicated in the photo. During the shuttle’s approach to Earth,
the glide angle changes. When the shuttle’s altitude is
about 15.7 miles, its horizontal distance to
the runway is about 59 miles. Find the
measure of the glide angle. Round
your answer to the nearest tenth.
Careers
UNITE
D ST
ATES
altitude
Not drawn
to scale
distance
to runway
runway
ASTRONAUTS on the space
shuttle include pilots qualified
to fly the shuttle and mission
specialists who conduct
scientific experiments in
space. All astronauts need to
have a strong background in
science and mathematics.
Inverse Sine and Inverse Cosine Find the measure of aA to the
nearest tenth of a degree.
34.
B
12
35. C
8
A
36. A
6
Career Links
C
A
CLASSZONE.COM
8
6
10
C
B
B
Solving Right Triangles Solve the right triangle. Round decimals to
the nearest tenth.
37. L
38.
17
M
16
K
sin
–1
39. T
P
N
You be the Judge
40.
P
4
14
S
6
15
R
Each of the expressions
A
30
AB
BC
BC
, cos–1 , and tan–1 can be used to
AC
AC
AB
19
approximate maA. Which expression would you
choose? Explain your choice.
Standardized Test
Practice
B
C
41. Multiple Choice Which additional information would not be
enough to solve T PQR?
A
C
maP and PR
PQ and PR
B maP and maR
D maP and PQ
P
P
R
KL
JL
L
J
JL
KL
K
42. Multiple Choice Which expression is correct?
574
Chapter 10
JL
JK
G
tan1 maJ
JL
JK
J
sin1 maK
F
sin1 maJ
H
cos1 maK
Right Triangles and Trigonometry
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Mixed Review
Circumference and Area of Circles Find the circumference and the
area of the circle. Round your answers to the nearest whole number.
(Lesson 8.7)
43.
44.
45.
15 in.
34 yd
8 cm
Volume of Solids Find the volume of the solid. Round your answers
to the nearest whole number. (Lesson 9.6)
46.
47.
48.
28 cm
10 ft
5 in.
Algebra Skills
Decimal Operations Evaluate. (Skills Review, p. 655)
49. 0.36 0.194
50. $8.42 $2.95
51. 7 4.65
52. 55.40 0.04
53. 700 0.35
54. $22.50 0.08
Quiz 2
Find the value of each variable. Round the results to the nearest
tenth. (Lessons 10.4, 10.5)
1.
2.
x
3.
2 63
6
40
y
42
x
12
x
4.
5.
19
21
y
6.
14
x
8
35
29
x
x
y
Use a calculator to approximate the value to four decimal places.
(Lessons 10.4, 10.5)
7. tan 72
8. sin 52
9. cos 36
Solve the right triangle. Round decimals to the nearest tenth.
(Lesson 10.6)
P
10.
11. P
q
m
16
R
12. J
8
P
7
3
K
k
L
12.4
40
M
q
N
10.6
Solving Right Triangles
575