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Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 163
Special Right Triangles5.1Goal p Use the relationships among the sides in
special right triangles.GeorgiaPerformanceStandard(s)
MM2G1a, MM2G1b
Your Notes
THEOREM 5.1: 458-458-908 TRIANGLE THEOREM
In a 458-458-908 triangle, the hypotenuse
2x
x
x458
458
is times as long as each leg.
hypotenuse 5 leg p
Find the value of x.
a.6
458
x
b. 29
x x
Solutiona. By the Triangle Sum Theorem, the measure of the
third angle must be . Then the triangle is a - -908 triangle, so by Theorem 5.1, the
hypotenuse is times as long as each leg.
hypotenuse 5 leg p - -908Triangle Theorem
x 5 Substitute.
b. You know that each of the two congruent angles in the triangle has a measure of because the sum of the angle measures in a triangle is 1808.
hypotenuse 5 leg p - -908Triangle Theorem
5 x p Substitute.
5 x Divide each side by .
5 x Simplify.
Example 1 Find lengths in a 458-458-908 triangle
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Your Notes
1.
22 22
x458
2.
212
xx
Checkpoint Find the value of x.
THEOREM 5.2: 308-608-908 TRIANGLE THEOREM
In a 308-608-908 triangle, the hypotenuse is as long as the shorter leg, and the longer leg is times as long as the shorter leg.
hypotenuse 5 p shorter leg x 2x
308
608
3xlonger leg 5 shorter leg p
Music You make a guitar pick that resembles an equilateral triangle with side lengths of 32 millimeters. What is the approximate height of the pick?
SolutionDraw the equilateral triangle described.
608
32 mmh32 mm
16 mm 16 mmDA C
B
Its altitude forms the longer leg of two - -908 triangles. The length
h of the altitude is approximately the height of the pick.
longer leg 5 shorter leg p
h 5 p < mm
Example 2 Find the height of an equilateral triangle
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Your Notes
Find the values of x and y. Write your answer in simplest radical form.
xy
308
608
8
SolutionStep 1 Find the value of x.
longer leg 5 shorter leg p
5 x Substitute.
5 x Divide each side by .
p 5 x Multiply numerator and denominator by .
5 x Multiply fractions.
Step 2 Find the value of y.
hypotenuse 5 p shorter leg
y 5 p 5
Example 3 Find lengths in a 308-608-908 triangle
Checkpoint Find the value of the variable.
3.
308
608
x
32
4.
h12 12
66
Homework
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166 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
Find the value of x. Write your answer in simplest form.
1.
24x
458
458
2.
x
3458
458
3. 28
xx
458 458
4.
27
x
x
5. 5
5x
6.
4
x
Complete the table.
7.
y
x
x
458
x 3 7 2
y 6 Ï}
2
LESSON
5.1 Practice
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Name ——————————————————————— Date ————————————
LESSON
5.1 Practice continued
8.
ac
b
60º
30º
a 4 6
b 5 Ï}
3
c 8 Ï}
3
Find the value of each variable. Write your answers in simplest form.
9.
x
y
60º
30º
6
10.
60º
30ºx
y 32
11. 60º
30ºx
y
9
12. 7
y 8 y 8
x
13.
xº
30º
y10
14. 60º
y
2x
36
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168 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
Find the value of each variable. Write your answers in simplest form.
15.
y
60º
xº
10
16.
2 y
y5x°
5x°
17.
2xº
xº
32
y
The side lengths of a triangle are given. Determine whether it is a 458-458-908 triangle, a 308-608-908 triangle, or neither.
18. 2, 4, 2 Ï}
3 19. 5, 5, 5 Ï}
2 20. 6, 12, 8
21. 11, 11, 11 Ï}
2 22. 10, 20, 10 Ï}
3 23. 3, 4, 5
LESSON
5.1 Practice continued
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LESSON
5.1 Practice continued
24. Construction A construction worker is building a ramp to
30°
8 ft
make transportation of materials easier between an upper and lower platform. The upper platform is 8 feet off the ground, and the angle of elevation is 308. How long is the ramp?
25. Art Gallery A designer is creating a new and unique
20 ft308
20 ft308
entryway for an art gallery. The designer wants the entryway to slant inward at a 308 angle on either side of the wall as shown.
a. How tall should the entryway be if the inner length of the entryway is 20 feet? Round your answer to the nearest foot.
b. How wide is the base of the entryway, from end to end?
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170 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
5.2 Apply the Tangent RatioGoal p Use the tangent ratio for indirect measurement.Georgia
PerformanceStandard(s)
MM2G2a, MM2G2b, MM2G2c
Your Notes
VOCABULARY
Trigonometry
Trigonometric ratio
Tangent
Complementary angles
TANGENT RATIO
Let nABC be a right triangle B
C A
with acute ∠A. The tangent of ∠A (written as tan A) is defined as follows:
tan A 5length of leg opposite ∠A}}}length of leg adjacent to ∠A 5
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Your Notes
Find the value of x.
25°
x
12
Use the tangent of an acute angle to find a leg length.
tan 258 5 Write ratio for tangent of 258.
tan 258 5 Substitute.
p tan 258 5 x Multiply each side by .
( ) ø x Use a calculator to find tan 258.
ø x Simplify.
Example 2 Find a leg length
1. Find tan A and tan B. Round
30
24 18
A B
C
to four decimal places.
2. Find the value of x. Round to
38°
x
20 AC
B
the nearest tenth.
Checkpoint Complete the following exercises.
Find tan X and tan Y. Write each Y
Z X
8
15
17answer as a fraction and as a decimal rounded to four places.
tan X 5opp. ∠X}adj. to ∠X 5 5 5
tan Y 5opp. ∠Y}adj. to ∠Y 5 5 5
In the right triangle, nXYZ, ∠X and ∠Y are angles. You can see that the
tangent ratios of the angles are .
Example 1 Find tangent ratios
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Your Notes
Find the height h of the flagpole
70°7 ft
h ft
to the nearest foot.
tan 708 5 Write ratio for tangent of 708.
tan 708 5 Substitute.
p tan 708 5 Multiply each side by .
ø h Use a calculator to simplify.
The flagpole is about feet tall.
Example 4 Estimate height using tangent
3. Find the height h of the flagpole in Example 4 to the nearest foot if the angle is 758.
Checkpoint Complete the following exercise.
Homework
Find tan X and tan Y for similar triangles. Then compare the tangent ratios.
c
a
b
Y
Z X
3c
3a
3b
Y
Z X
tan X 5 tan X 5 5
tan Y 5 tan Y 5 5
The values of tan X and tan Y for the similar triangles are .
Example 3 Compare the tangent ratios for similar triangles
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LESSON
5.2 PracticeFind tan A and tan B. Write each answer as a decimal rounded to four decimal places.
1. B C
A
1237
35 2. A
CB
29 21
20
3.
AC
B
3
4
5
Find the value of x to the nearest tenth.
4.
x458
15
5.
39°
24
x
6.
32°
9x
7.
34°12
x 8. 67°
18
x
9.
43°24
x
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Find tan X and tan Y for the similar triangles. Then compare the ratios.
10. Y
Z X24
2610
Y
Z X12
135
11.
XX
Y
Y
ZZ
33 65
56
13066
112
Find the value of x using the defi nition of tangent. Then fi nd the value of x using the 458-458-908 Triangle Theorem or the 308-608-908 Triangle Theorem. Compare the results.
12.
x
458
7
13. 1030°
x
14. 608
6 3
x
LESSON
5.2 Practice continued
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LESSON
5.2 Practice continued
Find the area of the triangle. Round your answer to the nearest tenth.
15.
x
48º17
16.
37°
x
24
17.
63°x
6
18.
53°
x 11
19.
27°
x
28
20.
73° x6
Use the tangent ratio to fi nd the value of x. Round to the nearest tenth.
21.
x
458
16 22.
x
428
35
23. x
57819
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24. Billboard You are standing 35 feet from a billboard.
58°35 ft
On
Sale Now!
The angle of elevation from your position to the top of the billboard is 588. How tall is the billboard? Round your answer to the nearest foot.
25. Tree Your friend is standing near a tree that is 18 feet tall.
37°
18 ft
Yourfriend
The angle of depression from the top of the tree to your friend is 378. How far is your friend from the tree? Round your answer to the nearest foot.
LESSON
5.2 Practice continued
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Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 177
Apply the Sine and Cosine Ratios
5.3Goal p Use the sine and cosine ratios.Georgia
PerformanceStandard(s)
MM2G2a, MM2G2b, MM2G2c
Your Notes
VOCABULARY
Sine
Cosine
SINE AND COSINE RATIOS
Let nABC be a right triangle with B
C A
acute ∠A. The sine of ∠A and cosine of ∠A (written sin A and cos A) are defined as follows:
sin A 5length of leg opposite ∠A}}}
length of hypotenuse 5
cos A 5length of leg adjacent to ∠A}}}
length of hypotenuse 5
Find sin X and sin Y. Write each Y
Z X
725
24
answer as a fraction and as a decimal rounded to four places.
sin X 5opp. ∠X}
hyp. 5 5 5
sin Y 5opp. ∠Y}
hyp. 5 5 5
Example 1 Find sine ratios
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Your Notes
Find cos X and cos Y. Write each Y
Z X
725
24
answer as a fraction and as a decimal rounded to four places.
cos X 5adj. to ∠X}
hyp. 5 5 5
cos Y 5adj. to ∠Y}
hyp. 5 5 5
Example 2 Find cosine ratios
1. Find sin A and sin B. B
C A
2129
20 2. Find cos A and cos B.
Checkpoint Find the indicated measure. Round to 4 decimal places, if necessary.
Use a trigonometric ratio to find the value of x in the diagram. Round to the nearest tenth.
a.x
31°12
b.
x
44°
48
a. cos 318 5 b. sin 448 5
cos 318 5 sin 448 5
x 5 p sin 448 5 x
x ø ( ) ø x
x ø ø x
Example 3 Use trigonometric ratios to find side lengths
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Your Notes
Find the height of the parking ramp shown.
sin 278 5 65 ft
27°x ft
sin 278 5
p sin 278 5 x
( ) ø x
ø x
Example 5 Use trigonometric ratios to find side lengths
3. Find the value of x. Round to
x
y
28
46°
the nearest tenth.
4. Find the value of y. Round to the nearest tenth.
Checkpoint Complete the following exercises.
Find the sine and cosine of Y
Z Xab c
M
N L3a
3b3c
∠X, ∠Y, ∠L, and ∠M of the similar triangles. Then compare the ratios.
sin X 5 cos X 5
sin Y 5 cos Y 5
sin L 5 5 cos L 5 5
sin M 5 5 cos M 5 5
In nXYZ, ∠X and ∠Y are angles, so sin X 5 cos and sin Y 5 cos . In nLMN, ∠L and ∠M are angles, so sin L 5 cos and sin M 5 cos . Because nXYZ and nLMN are
triangles, sin X 5 sin , cos X 5 cos ,sin Y 5 sin , and cos Y 5 cos .
Example 4 Sine and cosine ratios for similar triangles
Homework
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Find sin R and sin S. Write each answer as a decimal rounded to four decimal places.
1.
16 20
12
S
T R
2.
32 40
24T
R
S 3. 24
2610
S T
R
Find cos A and cos B. Write each answer as a decimal rounded to four decimal places.
4.
41
9
40
B
A
C
5.
6
B
AC
8 10
6.
39
80C A
B
89
7. Find sin X, sin Y, cos X, and cos Y. Then compare the ratios.
35
1237
XZ
Y
LESSON
5.3 Practice
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LESSON
5.3 Practice continued
8. Find sine and cosine of ∠ X, ∠ Y, ∠ L, and ∠ M. Then compare the ratios.
35
1237
XZ
Y
70
2474
N L
M
Use a sine or cosine ratio to fi nd the value of each variable. Round decimals to the nearest tenth.
9.
ab
24
72° 10. c
d18
63° 11.
r
s 32
52°
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Use a sine or cosine ratio to fi nd the value of each variable. Round decimals to the nearest tenth.
12.
b
a
24
42° 13.
c
d50
61° 14.
sr
1733°
Find the unknown side length. Then fi nd sin A and cos A. Write each answer as a decimal rounded to four decimal places.
15.
38
10c
A
BC
16.
9
12 c
A
B
C
17. 45
27a
A
B
C
LESSON
5.3 Practice continued
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LESSON
5.3 Practice continued
18. Car A car is driving down a hill that is 500 feet long,
32°
d
500 ft
Not drawn to scale
at an angle of elevation of 328. To the nearest foot, what is the vertical distance d covered by the car?
19. Satellite A satellite is launched at an angle of elevation
d
h
72°
Not drawn to scale
of 728 as shown.
a. Suppose the satellite covers a total distance of 8 miles during the launch. What is its vertical height h? Round to the nearest tenth of a mile.
b. The satellite reaches a vertical height of 3 miles. What is the total distance d covered during the launch? Round to the nearest tenth of a mile.
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5.4 Solve Right TrianglesGoal p Use inverse tangent, sine, and cosine ratios.Georgia
PerformanceStandard(s)
MM2G2c
Your Notes
VOCABULARY
Solve a right triangle
INVERSE TRIGONOMETRIC RATIOS
Let ∠A be an acute angle.
A
B
CInverse Tangent If tan A 5 x, then tan21 x 5 m∠A. tan21 BC
}AC 5 m∠A
Inverse Sine If sin A 5 y, then sin21 y 5 m∠A. sin21 BC
}AB 5 m∠A
Inverse Cosine If cos A 5 z, then cos21 z 5 m∠A. cos21 AC
}AB 5 m∠A
Use a calculator to approximate the
B
16
20
C
A
measure of ∠A to the nearest tenth of a degree.
Because tan A 5 5 5 ,
tan21 5 m∠A. Using a calculator, tan21 < .
So, the measure of ∠A is approximately .
Example 1 Use an inverse tangent to find an angle measure
1. In Example 1, use a calculator and an inverse tangent to approximate m∠C to the nearest tenth of a degree.
Checkpoint Complete the following exercise.
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Your Notes
Let ∠A and ∠B be acute angles in two right triangles. Use a calculator to approximate the measures of ∠A and ∠B to the nearest tenth of a degree.
a. sin A 5 0.76 b. cos B 5 0.17
Solution
a. m∠A 5 b. m∠B 5
< <
Example 2 Use an inverse sine and an inverse cosine
Solve the right triangle.
238
A C
B
40 ftRound decimal answers to the nearest tenth.
SolutionStep 1 Find m∠B by using the Triangle Sum Theorem.
5 908 1 238 1 m∠B
5 m∠B
Step 2 Approximate BC using a ratio.
5BC}40 Write ratio for .
5 BC Multiply each side by .
< BC Approximate .
< BC Simplify and round answer.
Step 3 Approximate AC using a ratio.
5AC}40 Write ratio for .
5 AC Multiply each side by .
< AC Approximate .
< AC Simplify and round answer.
The angle measures are , , and . The side lengths are feet, about feet, and about
feet.
Example 3 Solve a right triangle
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Your Notes
Homework
2. Find m∠T to the nearest tenth of a degree if cos T 5 0.64.
3. Find m∠D to the nearest tenth of a degree if sin D 5 0.48.
4. Solve a right triangle that has a 508 angle and a 15 inch hypotenuse.
Checkpoint Complete the following exercises.
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LESSON
5.4 PracticeMatch the trigonometric expression with the correct ratio. Some ratios may be used more than once, and some may not be used at all.
1. sin A 2. cos A 3. tan A
A C
B
8
15
17
4. sin B 5. cos B 6. tan B
A. 8 }
17 B.
15 }
17 C.
17 }
8
D. 17
} 15
E. 8 }
15 F.
15 }
8
Use a calculator to approximate the measure of ∠ A to the nearest tenth of a degree.
7. A
B C30
20
8.
A
B C15
11
9. A B
C
1426
10. C B
A
1016
11. A
B C7
11
12. B C
A
9 22
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Solve the right triangle. Round decimal answers to the nearest tenth.
13.
R
P
12
458
14.
P
N
12
17 15. U
S
T
15708
16. V M
D
21
508
17.
R
T
A
16
30
18. EU
M
15
18
LESSON
5.4 Practice continued
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LESSON
5.4 Practice continued
Let ∠ A be an acute angle in a right triangle. Approximate the measure of ∠ A to the nearest tenth of a degree.
19. sin A 5 0.45 20. tan A 5 0.9 21. sin A 5 0.76 22. cos A 5 0.32
23. tan A 5 5.2 24. cos A 5 0.24 25. sin A 5 0.15 26. cos A 5 0.66
27. Multiple Choice Using the diagram to the right, for what
xA C
Bvalue of x does sin A 5 cos A?
A. 30° B. 45°
C. 60° D. none
28. Ladder You lean a 20 foot ladder against a house. The base
20 ft
4 ftu
of the ladder is 4 feet from the wall. What angle u does the ladder make with the ground?
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29. Skyscraper You are standing 350 feet away
x 8
350 ft
750 ft
from a skyscraper that is 750 feet tall. What is the angle of elevation from you to the top of the building?
30. Concert You attend a music concert with some
248
stage
45 ftd
friends and sit halfway up the bleachers in the arena. The angle of depression from your horizontal line of sight to the stage is 24°. If your seat is 45 feet above stage level, what is your actual distance d from the stage? Round to the nearest foot.
LESSON
5.4 Practice continued
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Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 191
Words to ReviewGive an example of the vocabulary word.
Trigonometry
Tangent
Cosine
Solve a right triangle
Inverse sine
Trigonometric ratio
Complementary angles
Sine
Inverse tangent
Inverse cosine
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