Download 7.6 Apply the Sine and Cosine Ratios

7.6
Apply the Sine and Cosine
Ratios
Goal
Your Notes
p Use the sine and cosine ratios.
VOCABULARY
Sine, cosine
Angle of elevation
Angle of depression
SINE AND COSINE RATIOS
C
Let nABC be a right triangle
leg
with acute ∠A. The sine of ∠A opposite
aA
and cosine of ∠A (written sin A
B
and cos A) are defined as follows:
Remember these
abbreviations:
sine → sin
cosine → cos
hypotenuse → hyp
hypotenuse
leg adjacent to aA A
length of leg opposite ∠A
5
sin A 5 }}}
length of hypotenuse
length of leg adjacent to ∠A
5
cos A 5 }}}
length of hypotenuse
Copyright © Holt McDougal. All rights reserved.
Lesson 7.6 • Geometry Notetaking Guide
195
7.6
Apply the Sine and Cosine
Ratios
Goal
Your Notes
p Use the sine and cosine ratios.
VOCABULARY
Sine, cosine Sine and cosine are trigonometric
ratios for acute angles that involve the lengths
of a leg and the hypotenuse of a right triangle.
Angle of elevation When looking up at an object,
the angle your line of sight makes with a horizontal
line is called the angle of elevation.
Angle of depression When looking down at an
object, the angle your line of sight makes with a
horizontal line is called the angle of depression.
SINE AND COSINE RATIOS
C
Let nABC be a right triangle
leg
with acute ∠A. The sine of ∠A opposite
aA
and cosine of ∠A (written sin A
B
and cos A) are defined as follows:
Remember these
abbreviations:
sine → sin
cosine → cos
hypotenuse → hyp
length of leg opposite ∠A
5
sin A 5 }}}
length of hypotenuse
length of leg adjacent to ∠A
leg adjacent to aA A
BC
AC
5
cos A 5 }}}
length of hypotenuse
Copyright © Holt McDougal. All rights reserved.
hypotenuse
AB
AC
Lesson 7.6 • Geometry Notetaking Guide
195
Your Notes
Find sine ratios
Example 1
Find sin U and sin W. Write each answer
as a fraction and as a decimal rounded
to four places.
U
34
30
Solution
opp. ∠U
5
sin U 5 }
hyp.
W
5
opp. ∠W
sin W 5 }
5
hyp.
5
5
5
16
V
<
<
Find cosine ratios
Example 2
Find cos S and cos R. Write each answer
as a fraction and as a decimal rounded to
four places.
S
53
45
Solution
adj. to ∠S
cos S 5 }
hyp.
adj. to ∠R
cos R 5 }
hyp.
R
5
<
5
<
28
T
Checkpoint Find sin B, sin C, cos B, and cos C. Write
each answer as a fraction and as a decimal rounded
to four places.
1.
A
20
21
C
29
B
196 Lesson 7.6 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Find sine ratios
Example 1
Find sin U and sin W. Write each answer
as a fraction and as a decimal rounded
to four places.
U
34
30
Solution
opp. ∠U
5
sin U 5 }
hyp.
opp. ∠W
sin W 5 }
5
hyp.
WV
UW
5
UV
5
UW
16
34
30
34
5
5
8
17
W
16
V
< 0.4706
15
17
< 0.8824
Find cosine ratios
Example 2
Find cos S and cos R. Write each answer
as a fraction and as a decimal rounded to
four places.
S
53
45
Solution
adj. to ∠S
cos S 5 }
hyp.
adj. to ∠R
cos R 5 }
hyp.
ST
SR
RT
SR
5
5
45
53
28
53
R
28
T
< 0.8491
< 0.5283
Checkpoint Find sin B, sin C, cos B, and cos C. Write
each answer as a fraction and as a decimal rounded
to four places.
1.
A
20
21
C
29
B
20
21
< 0.6897,
< 0.7241, sin C 5 }
sin B 5 }
29
29
20
21
< 0.7241
< 0.6897, cos C 5 }
cos B 5 }
29
29
196 Lesson 7.6 • Geometry Notetaking Guide
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Your Notes
Example 3
Use a trigonometric ratio to find a hypotenuse
Basketball You walk from one corner of
a basketball court to the opposite corner.
Write and solve a proportion using a
trigonometric ratio to approximate the
distance of the walk.
x ft
628
Solution
sin 628 5
Write ratio for sine of 628.
sin 628 5
Substitute.
p
5
Multiply each side by
Example 4
Divide each side by
x<
Use a calculator to find
.
x<
Simplify.
.
feet.
Find a hypotenuse using an angle of depression
Roller Coaster You are at the top of
a roller coaster 100 feet above the
ground. The angle of depression is
448. About how far do you ride down
the hill?
448
x ft
100 ft
sin 448 5
Write ratio for sine of 448.
sin 448 5
Substitute.
5
Multiply each side by
x5
Divide each side by
x<
Use a calculator to find
.
x<
Simplify.
You ride about
Copyright © Holt McDougal. All rights reserved.
.
x5
The distance of the walk is about
xp
94 ft
.
.
feet down the hill.
Lesson 7.6 • Geometry Notetaking Guide
197
Your Notes
Example 3
Use a trigonometric ratio to find a hypotenuse
Basketball You walk from one corner of
a basketball court to the opposite corner.
Write and solve a proportion using a
trigonometric ratio to approximate the
distance of the walk.
x ft
94 ft
628
Solution
opp.
hyp.
Write ratio for sine of 628.
94
x
Substitute.
sin 628 5 }
sin 628 5 }
x p sin 628 5 94
Multiply each side by x .
94
sin 628
Divide each side by sin 628 .
x< }
94
0.8829
Use a calculator to find
sin 628 .
x < 106.5
Simplify.
x5 }
The distance of the walk is about 106.5 feet.
Example 4
Find a hypotenuse using an angle of depression
Roller Coaster You are at the top of
a roller coaster 100 feet above the
ground. The angle of depression is
448. About how far do you ride down
the hill?
opp.
hyp.
sin 448 5 }
100
sin 448 5 }
x
x p sin 448 5 100
100
sin 448
x5 }
100
0.6947
448
x ft
100 ft
Write ratio for sine of 448.
Substitute.
Multiply each side by x .
Divide each side by sin 448 .
x< }
Use a calculator to find
sin 448 .
x < 143.9
Simplify.
You ride about 144 feet down the hill.
Copyright © Holt McDougal. All rights reserved.
Lesson 7.6 • Geometry Notetaking Guide
197
Your Notes
Checkpoint Complete the following exercises.
2. In Example 3, use the cosine ratio to approximate
the width of the basketball court.
3. Suppose the angle of depression in Example 4 is
728. About how far would you ride down the hill?
Example 5
Find leg lengths using an angle of elevation
Railroad A railroad crossing arm
that is 20 feet long is stuck with
an angle of elevation of 358.
Find the lengths x and y.
20 ft
x ft
358
y ft
Solution
Step 1 Find x.
opp.
5}
hyp.
Write ratio for
of
.
5
Substitute.
5x
Multiply each side by
<x
Use a calculator to simplify.
.
Step 2 Find y.
adj.
198 Lesson 7.6 • Geometry Notetaking Guide
5}
hyp.
Write ratio for
of
.
5
Substitute.
5y
Multiply each side by
<y
Use a calculator to simplify.
.
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Complete the following exercises.
2. In Example 3, use the cosine ratio to approximate
the width of the basketball court.
about 50 feet
3. Suppose the angle of depression in Example 4 is
728. About how far would you ride down the hill?
about 105 feet
Example 5
Find leg lengths using an angle of elevation
Railroad A railroad crossing arm
that is 20 feet long is stuck with
an angle of elevation of 358.
Find the lengths x and y.
20 ft
x ft
358
y ft
Solution
Step 1 Find x.
opp.
sin 358 5 }
hyp.
Write ratio for
of 358 .
x
sin 358 5 }
Substitute.
20
sine
20 p sin 358 5 x
Multiply each side by 20 .
11.5 < x
Use a calculator to simplify.
Step 2 Find y.
adj.
cos 358 5 }
hyp.
y
20
cos 358 5 }
198 Lesson 7.6 • Geometry Notetaking Guide
Write ratio for cosine
of 358 .
Substitute.
20 p cos 358 5 y
Multiply each side by 20 .
16.4 < y
Use a calculator to simplify.
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 6
Use a special right triangle to find a sin and cos
Use a special right triangle to find the sine and cosine
of a 308 angle.
Solution
Use the 308-608-908 Triangle Theorem
to draw a right
}
. Then set up
triangle with side lengths of 1, Ï3 , and
sine and cosine ratios for the 308 angle.
sin 308 5
5
cos 308 5
5
3
5
308
<
1
2
608
Checkpoint Complete the following exercises.
4. In Example 5, suppose the angle of elevation is 408.
What are the new lengths x and y ?
5. Use a special right triangle to find the sine and
cosine of a 608 angle.
Homework
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Lesson 7.6 • Geometry Notetaking Guide
199
Your Notes
Example 6
Use a special right triangle to find a sin and cos
Use a special right triangle to find the sine and cosine
of a 308 angle.
Solution
Use the 308-608-908 Triangle Theorem
to draw a right
}
triangle with side lengths of 1, Ï3 , and 2 . Then set up
sine and cosine ratios for the 308 angle.
opp.
1
sin 308 5 } 5 } 5 0.5000
hyp.
3
2
}
adj.
Ï3
cos 308 5 } 5 } < 0.8660
2
hyp.
308
1
2
608
Checkpoint Complete the following exercises.
4. In Example 5, suppose the angle of elevation is 408.
What are the new lengths x and y ?
x < 12.9, y < 15.3
5. Use a special right triangle to find the sine and
cosine of a 608 angle.
sin 608 < 0.8660
cos 608 5 0.5000
Homework
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Lesson 7.6 • Geometry Notetaking Guide
199
Focus On
Trig
Use after Lesson 7.6
Your Notes
Cotangent, Secant, and
Cosecant Ratios
Goal
p Use the cotangent, secant, and cosecant ratios.
VOCABULARY
Cotangent, secant, cosecant
PROPERTY SUMMARY BOX
Let nABC be a right triangle with acute /A. The
cotangent of /A, secant of /A, and cosecant of /A
(written as cot A, sec A, and csc A) are:
/A
length of leg
cot A 5 }}}
/A
length of leg
length of
sec A 5 }}}
/A
length of leg
Remember these
abbreviations:
cot
cotangent
sec
secant
csc
cosecant
length of
csc A 5 }}}
/A
length of leg
Note that these ratios are the
reciprocals of the tangent, cosine,
and sine ratios.
1
cot A 5 }
Example 1
5}
5}
5}
B
leg
opposite
A
C
hypotenuse
A
leg adjacent to A
1
1
sec A 5 } csc A 5 }
Find sec F.
Find sec F.
F
hyp.
sec F 5 } 5 } 5
adj. to /F
26
10
D
200 7.6 Focus on Trigonometry • Geometry Notetaking Guide
24
E
Copyright © Holt McDougal. All rights reserved.
Focus On
Trig
Use after Lesson 7.6
Your Notes
Cotangent, Secant, and
Cosecant Ratios
Goal
p Use the cotangent, secant, and cosecant ratios.
VOCABULARY
Cotangent, secant, cosecant Cotangent, secant, and
cosecant are trigonometric ratios for acute angles
involving the side lengths of a right triangle.
PROPERTY SUMMARY BOX
Let nABC be a right triangle with acute /A. The
cotangent of /A, secant of /A, and cosecant of /A
(written as cot A, sec A, and csc A) are:
AC
length of leg adjacent to /A
}}}
5}
cot A 5
BC
opposite /A
length of leg
Remember these
abbreviations:
cot
cotangent
sec
secant
csc
cosecant
length of hypotenuse
sec A 5 }}}
length of leg adjacent to /A
AB
5}
AC
length of hypotenuse
csc A 5 }}}
opposite /A
length of leg
AB
5}
BC
Note that these ratios are the
reciprocals of the tangent, cosine,
and sine ratios.
1
cot A 5 }
tan A
Example 1
B
leg
opposite
A
C
hypotenuse
A
leg adjacent to A
1
1
sec A 5 } csc A 5 }
cos A
sin A
Find sec F.
Find sec F.
F
26
hyp.
sec F 5 } 5 } 5 2.6
adj. to /F
10
200 7.6 Focus on Trigonometry • Geometry Notetaking Guide
26
10
D
24
E
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Your Notes
Checkpoint Find cot O and csc O.
1.
F
17
8
O
N
15
Use trigonometric ratios
Example 2
Find the value of the variable.
a.
b.
x
7
4.2
15°
63°
l
Solution
If your calculator
does not have a csc
key, you can find
sin 158 and use the
x21 key to find
csc 158.
You are given an angle,
a side
it, and
asked to find the length
of the hypotenuse.
You are given an angle,
a side
it, and
asked to find the length
of the side
to it.
Use cosecant 5 }.
Use cotangent 5 }.
5 }.
csc
5 }.
cot
csc 158 5 }
cot 638 5 }
5x
5I
≈x
≈I
Checkpoint Find the value of the variables.
Homework
2.
36°
n
29
m
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7.6 Focus on Trigonometry • Geometry Notetaking Guide
201
Your Notes
Checkpoint Find cot O and csc O.
1.
15
cot O 5 }
5 1.875,
F
8
17
csc O 5 } = 2.125
8
17
8
O
N
15
Use trigonometric ratios
Example 2
Find the value of the variable.
a.
b.
x
7
4.2
15°
63°
l
Solution
You are given an angle,
a side opposite it, and
asked to find the length
of the hypotenuse.
hyp.
}
Use cosecant 5
.
opp.
If your calculator
does not have a csc
key, you can find
sin 158 and use the
x21 key to find
csc 158.
hyp.
csc 158 5 }.
opp.
x
You are given an angle,
a side opposite it, and
asked to find the length
of the side adjacent to it.
adj.
}
Use cotangent 5
.
opp.
adj.
cot 638 5 }.
opp.
l
csc 158 5 }
7
cot 638 5 }
7 csc 158 5 x
4.2 cot 638 5 I
27.05 ≈ x
2.14 ≈ I
4.2
Checkpoint Find the value of the variables.
Homework
2.
36°
n
29
hyp.
adj.
cot 5 }
opp.
n
sec 368 5 }
29 sec 368 = n
cot 368 5 29
}
m
29
m5 }
35.85 ≈ n
21.07 ≈ m
29
m
Copyright © Holt McDougal. All rights reserved.
adj.
sec 5 }
cot 368
7.6 Focus on Trigonometry • Geometry Notetaking Guide
201