Download Chapter Summary and Review

Page 1 of 4
10
Chapter
Summary and Review
VOCABULARY
• radical, p. 537
• leg opposite an angle, p. 557
• solve a right triangle, p. 569
• radicand, p. 537
• leg adjacent to an angle, p. 557
• inverse tangent, p. 569
• 458-458-908 triangle, p. 542
• tangent, p. 557
• inverse sine, p. 570
• 308-608-908 triangle, p. 549
• sine, p. 563
• inverse cosine, p. 570
• trigonometric ratio, p. 557
• cosine, p. 563
VOCABULARY REVIEW
Fill in the blank.
1. A(n) __?__ is an expression written with a radical symbol.
2. The number or expression inside the radical symbol is the __?__.
3. A(n) __?__ is a ratio of the lengths of two sides of a right triangle.
4. A right triangle with side lengths 9, 9Ï3
w, and 18 is a(n) __?__ triangle.
5. To __?__ a right triangle means to determine the lengths of all three
sides of the triangle and the measures of both acute angles.
w is a(n) __?__ triangle.
6. A right triangle with side lengths 4, 4, and 4Ï2
7. If aF is an acute angle of a right triangle, then
leg opposite aF
leg adjacent to aF
__?__ of aF 5 }}}.
8. If aF is an acute angle of a right triangle, then
leg adjacent aF
hypotenuse
__?__ of aF 5 }}.
9. If aF is an acute angle of a right triangle, then
leg opposite aF
hypotenuse
__?__ of aF 5 }}.
10.1
EXAMPLES
Use the Product Property of Radicals to simplify the expression.
a. Ï7
w p Ï5
w
Ï7
w p Ï5
w 5 Ï7
wwpw
5 5 Ï3
w5
w
576
Examples on
pp. 537–538
SIMPLIFYING SQUARE ROOTS
Chapter 10
Right Triangles and Trigonometry
b. Ï5
w2
w
Ï5
w2
w 5 Ï4
wwpw3
1w 5 2Ï1
w3
w
Page 2 of 4
Chapter Summary and Review continued
Multiply the radicals. Then simplify if possible.
10. Ï1
w3
w p Ï1
w3
w
11. Ï2
w p Ï7
w2
w
12. Ï7
w p Ï1
w0
w
14. Ï3
w p Ï1
w9
w
15. Ï5
w p Ï5
w
16. Ï6
w p Ï1
w8
w
( )
17. (3Ï1
w1
w)2
13. 4Ï7
w 2
Simplify the radical expression.
18. Ï2
w7
w
19. Ï7
w2
w
20. Ï1
w5
w0
w
21. Ï6
w8
w
22. Ï1
w0
w8
w
23. Ï8
w0
w
24. Ï7
w5
w0
w0
w
25. Ï5
w0
w7
w
Use the formula A 5 lw to find the area of the rectangle. Round your
answer to the nearest tenth.
26.
27.
28.
3Ï·
3
2Ï·5
4Ï·
6
5Ï·2
10.2
Ï·
1·0
2Ï·
2
Examples on
pp. 542–544
458-458-908 TRIANGLES
EXAMPLES
Find the value of x.
a.
b.
71 458
x
458
x
458
33Ï·
2
458
By the 458-458-908 Triangle Theorem,
the length of the hypotenuse is the
length of a leg times Ï2
w, so x 5 71Ï2
w.
By the 458-458-908 Triangle Theorem,
the length of the hypotenuse is the
length of a leg times Ï2
w, so 33Ï2
w 5 x Ï2
w,
and x 5 33.
Find the length of the hypotenuse in the 458-458-908 triangle. Write
your answer in radical form.
29.
30.
x
458
15
31.
5Ï·
2
x
458
458
Ï·
7
458
x
458
458
Find the length of each leg in each 458-458-908 triangle. Write your
answer in radical form or as a decimal to the nearest tenth.
32.
x
458
19Ï·2
33.
458 x
34.
3Ï·
2
458
x
458
x
x
10
x
Chapter Summary and Review
577
Page 3 of 4
Chapter Summary and Review continued
10.3
Examples on
pp. 549–551
308-608-908 TRIANGLES
EXAMPLES
Find the value of each variable.
a.
b.
57
30
x
608
308
x
308
y
By the 308-608-908 Triangle Theorem,
the length of the hypotenuse is twice
the length of the shorter leg, so
x 5 2(57) 5 114.
The length of the longer leg is the
w,
length of the shorter leg times Ï3
so y 5 57Ï3
w.
y
608
By the 308-608-908 Triangle Theorem,
the length of the longer leg is the length
w,
of the shorter leg times Ï3
30
Ï3
w
so 30 5 xÏ3
w and x 5 } ≈ 17.3.
Then y 5 2x ≈ 34.6.
Find the value of each variable. Write your answers in radical form or
as a decimal to the nearest tenth.
35.
36.
37.
x
y 308
x 308 y
608
25
10.4
x
608
38.
y
x 608
308
3
94Ï·
45
308 y
608
19
Examples on
pp. 557–559
TANGENT RATIO
EXAMPLE
Find tan A and tan B as fractions and as decimals.
leg opposite to a A
leg adjacent to a A
21
20
A
tan A 5 }}} 5 }} 5 1.05
29
leg opposite to a B
20
tan B 5 }}} 5 }} ≈ 0.9524
21
leg adjacent to a B
B
20
C
21
Find tan A and tan B as fractions in simplest form and as decimals.
Round your answers to four decimal places if necessary.
B
39.
40. A
3
20Ï·
C
41. A
34
16
40
A
30
C
20
8
B
C
1·3
4Ï·
12
B
Approximate the value to four decimal places.
42. tan 178
578
Chapter 10
43. tan 818
Right Triangles and Trigonometry
44. tan 368
45. tan 248
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Chapter Summary and Review continued
10.5
Examples on
pp. 563–565
SINE AND COSINE RATIOS
EXAMPLE
Find sin A and cos A as fractions and as decimals.
leg opposite a A
hypotenuse
B
20
29
sin A 5 }} 5 }} ≈ 0.6897
leg adjacent to a A
hypotenuse
29
21
29
A
cos A 5 }}} 5 }} ≈ 0.7241
20
C
21
Find sin A and cos A as fractions in simplest form and as decimals.
Round your answers to four decimal places if necessary.
46.
47. C
B
25
1·4
2Ï·
C
15
20
30Ï·2
30Ï·
2
5
A
48.
A
C
9
B
A
60
B
Approximate the value to four decimal places.
49. sin 578
50. sin 128
51. cos 318
52. cos 758
53. Find the lengths of the legs of the triangle. Round your
10
answers to the nearest tenth.
x
318
y
10.6
Examples on
pp. 569–571
SOLVING RIGHT TRIANGLES
EXAMPLE
Solve T XYZ.
By the Pythagorean Theorem, y 2 5 52 1 72 5 25 1 49 5 74, so y ≈ 8.6.
X
7
tan X 5 }} 5 1.4, so maX 5 tan21 1.4 ≈ 54.58.
5
5
Since aX and aZ are complementary,
maZ 5 908 2 maX ≈ 908 – 54.58 5 35.58.
Y
y
7
Z
aA is an acute angle. Use a calculator to approximate the measure of
aA to the nearest tenth of a degree.
54. tan A 5 3.2145
55. sin A 5 0.0888
56. cos A 5 0.2243
57. tan A 5 1.2067
Solve the right triangle. Round decimals to the nearest tenth.
58.
B
c
q
59. P
3
14
A
6
R
C
23
P
60. Y
Z
x
z
558
3
X
Chapter Summary and Review
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