Download Apply the Tangent Ratio

7.5
Apply the Tangent Ratio
Goal
Your Notes
p Use the tangent ratio for indirect measurement.
VOCABULARY
Trigonometric ratio
Tangent
TANGENT RATIO
Remember these
abbreviations:
tangent → tan
opposite → opp.
adjacent → adj.
Let nABC be a right triangle
with acute ∠A. The tangent
of ∠A (written as tan A) is
defined as follows:
C
leg
opposite
aA
hypotenuse
B leg adjacent to aA A
length of leg opposite ∠A
5
tan A 5 }}}
length of leg adjacent to ∠A
Example 1
Find tangent ratios
Find tan S and tan R. Write each answer as a fraction
and as a decimal rounded to four places, if necessary.
S
65
25
T
Unless told
otherwise,
round values of
trigonometric
ratios to the
ten-thousandths’
place and round
lengths to the
tenths’ place.
192
60
R
Solution
opp. ∠S
adj. to ∠S
tan S 5 } 5
opp. ∠R
5
tan R 5 }
adj. to ∠R
Lesson 7.5 • Geometry Notetaking Guide
5
5
5
5
5
<
Copyright © Holt McDougal. All rights reserved.
7.5
Apply the Tangent Ratio
Goal
Your Notes
p Use the tangent ratio for indirect measurement.
VOCABULARY
Trigonometric ratio A trigonometric ratio is a ratio of
the lengths of two sides in a right triangle.
Tangent The ratio of the length of the leg opposite
an acute angle in a right triangle to the length of
the leg adjacent to the angle is the tangent of the
angle.
TANGENT RATIO
Remember these
abbreviations:
tangent → tan
opposite → opp.
adjacent → adj.
Let nABC be a right triangle
with acute ∠A. The tangent
of ∠A (written as tan A) is
defined as follows:
C
leg
opposite
aA
B leg adjacent to aA A
length of leg opposite ∠A
BC
5
tan A 5 }}}
length of leg adjacent to ∠A
Example 1
hypotenuse
AB
Find tangent ratios
Find tan S and tan R. Write each answer as a fraction
and as a decimal rounded to four places, if necessary.
S
65
25
T
Unless told
otherwise,
round values of
trigonometric
ratios to the
ten-thousandths’
place and round
lengths to the
tenths’ place.
192
60
R
Solution
opp. ∠S
adj. to ∠S
tan S 5 } 5
opp. ∠R
5
tan R 5 }
adj. to ∠R
Lesson 7.5 • Geometry Notetaking Guide
RT
5
ST
ST
RT
60
5
25
5
25
60
12
5 2.4
5
5
5
< 0.4167
12
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Find tan B and tan C. Write each answer
as a fraction and as a decimal rounded to four places.
1.
48
A
14
C
50
B
Example 2
Find a leg length
Find the value of x.
Use the tangent of an acute
angle to find a leg length.
17
318
x
tan 318 5
Write ratio for tangent of 318.
tan 318 5
Substitute.
p tan 318 5
Example 3
Multiply each side by
x5
Divide each side by
x<
Use a calculator to find
.
x<
Simplify.
.
.
Estimate height using tangent
Lighthouse Find the height h of the
lighthouse to the nearest foot.
opp.
p
Copyright © Holt McDougal. All rights reserved.
5}
adj.
Write ratio for
.
5
Substitute.
5h
Multiply each side by
<h
Use a calculator and simplify.
h
628
100 ft
Lesson 7.5 • Geometry Notetaking Guide
.
193
Your Notes
Checkpoint Find tan B and tan C. Write each answer
as a fraction and as a decimal rounded to four places.
1.
48
A
14
C
50
B
24
7
tan B 5 }
< 3.4286, tan C 5 }
< 0.2917
7
Example 2
24
Find a leg length
Find the value of x.
Use the tangent of an acute
angle to find a leg length.
17
318
x
opp.
tan 318 5 }
adj.
Write ratio for tangent of 318.
17
tan 318 5 }
Substitute.
x
x p tan 318 5 17
Multiply each side by x .
17
x5 }
Divide each side by tan 318 .
17
x< }
Use a calculator to find
tan 318 .
x < 28.3
Simplify.
tan 318
0.6009
Example 3
Estimate height using tangent
Lighthouse Find the height h of the
lighthouse to the nearest foot.
opp.
tan 628 5 }
adj.
Write ratio for
tan 628 .
h
tan 628 5 }
Substitute.
100
h
628
100 ft
100 p tan 628 5 h
Multiply each side by 100 .
188 < h
Use a calculator and simplify.
Copyright © Holt McDougal. All rights reserved.
Lesson 7.5 • Geometry Notetaking Guide
193
Your Notes
Use a special right triangle to find a tangent
Example 4
Use a special right triangle to find the tangent of a
308 angle.
Solution
as the length of the shorter leg to
Step 1 Choose
simplify calculations. Use the 308-608-908 Triangle
Theorem to find the length of the longer leg.
3
308
x
longer leg 5
5
x5
Step 2 Find tan 308.
The tangents of
all 308 angles are
the same constant
ratio. Any right
triangle with a 308
angle can be used
to determine this
value.
tan 308 5
Write ratio for tangent of 308.
tan 308 5
Substitute.
The tangent of any 308 angle is
<
.
Checkpoint In Exercises 2 and 3, find the value of x.
Round to the nearest tenth.
2.
x
3.
638
x
21
598
13
4. In Example 4, suppose
the length of the shorter leg
}
is 1 instead of }Ï 3 . Show that the tangent of 308 is
Homework
Ï3
.
still equal to }
3
194 Lesson 7.5 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Use a special right triangle to find a tangent
Example 4
Use a special right triangle to find the tangent of a
308 angle.
Solution
}
Step 1 Choose Ï3 as the length of the shorter leg to
simplify calculations. Use the 308-608-908 Triangle
Theorem to find the length of the longer leg.
3
308
x
}
longer leg 5 shorter leg p Ï3
}
}
x 5 Ï3 p Ï3 5 3
Step 2 Find tan 308.
The tangents of
all 308 angles are
the same constant
ratio. Any right
triangle with a 308
angle can be used
to determine this
value.
opp.
adj.
tan 308 5 }
Write ratio for tangent of 308.
}
Ï3
tan 308 5 }
3
Substitute.
}
Ï3
The tangent of any 308 angle is } < 0.5774 .
3
Checkpoint In Exercises 2 and 3, find the value of x.
Round to the nearest tenth.
2.
x
3.
638
x
21
598
13
x < 6.6
x < 34.9
4. In Example 4, suppose
the length of the shorter leg
}
is 1 instead of }Ï 3 . Show that the tangent of 308 is
Homework
Ï3
.
still equal to }
3
}
longer leg 5 shorter leg p Î3
}
}
x 5 1 p Ï3 5 Ï3
}
}
opp.
Ï3
Ï3
1
1
p}
tan 308 5 } 5 }
} 5 }
}
} 5 }
3
adj.
Ï3
Ï3
Ï3
194 Lesson 7.5 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.