Download 5.2 Right Triangle Trigonometry

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Section 5.2 Right Triangle Trigonometry
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Critical Thinking Exercises
Preview Exercises
Make Sense? In Exercises 117–120, determine whether
each statement makes sense or does not make sense, and explain
your reasoning.
Exercises 124–126 will help you prepare for the material covered
in the next section. In each exercise, let u be an acute angle in a right
triangle, as shown in the figure. These exercises require the use of
the Pythagorean Theorem.
117. I made an error because the angle I drew in standard
position exceeded a straight angle.
Length of the
hypotenuse
118. When an angle’s measure is given in terms of p, I know that
it’s measured using radians.
c
119. When I convert degrees to radians, I multiply by 1, choosing
p
180° for 1.
120. Using radian measure, I can always find a positive angle
less than 2p coterminal with a given angle by adding or
subtracting 2p.
121. If u = 32 , is this angle larger or smaller than a right angle?
122. A railroad curve is laid out on a circle. What radius should be
used if the track is to change direction by 20° in a distance of
100 miles? Round your answer to the nearest mile.
123. Assuming Earth to be a sphere of radius 4000 miles, how
many miles north of the Equator is Miami, Florida, if it is
26° north from the Equator? Round your answer to the
nearest mile.
Section
5.2
B
a
Length of the
side opposite u
u
A
C
b
Length of the side adjacent to u
124. If a = 5 and b = 12, find the ratio of the length of the side
opposite u to the length of the hypotenuse.
125. If a = 1 and b = 1, find the ratio of the length of the side
opposite u to the length of the hypotenuse. Simplify the ratio
by rationalizing the denominator.
a 2
b 2
126. Simplify: a b + a b .
c
c
Right Triangle Trigonometry
Objectives
� Use right triangles to evaluate
�
�
�
�
�
trigonometric functions.
Find function values for
p
p
30° a b, 45° a b, and
6
4
p
60° a b.
3
Recognize and use
fundamental identities.
Use equal cofunctions of
complements.
Evaluate trigonometric
functions with a calculator.
Use right triangle
trigonometry to solve applied
problems.
In the last century, Ang Rita Sherpa
climbed Mount Everest ten times, all
without the use of bottled oxygen.
M
ountain climbers have
forever been fascinated
by reaching the top of
Mount Everest, sometimes
with tragic results. The
mountain, on Asia’s TibetNepal border, is Earth’s highest, peaking at an incredible 29,035 feet. The heights
of mountains can be found using trigonometry. The word “trigonometry” means
“measurement of triangles.” Trigonometry is used in navigation, building, and
engineering. For centuries, Muslims used trigonometry and the stars to navigate
across the Arabian desert to Mecca, the birthplace of the prophet Muhammad,
the founder of Islam. The ancient Greeks used trigonometry to record the
locations of thousands of stars and worked out the motion of the Moon relative
to Earth. Today, trigonometry is used to study the structure of DNA, the master
molecule that determines how we grow from a single cell to a complex, fully
developed adult.
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498 Chapter 5 Trigonometric Functions
�
Use right triangles to evaluate
trigonometric functions.
Side opposite u
Hypotenuse
u
Side adjacent to u
Figure 5.19 Naming a right triangle’s
The Six Trigonometric Functions
We begin the study of trigonometry by defining six functions, the six trigonometric
functions. The inputs for these functions are measures of acute angles in right
triangles. The outputs are the ratios of the lengths of the sides of right triangles.
Figure 5.19 shows a right triangle with one of its acute angles labeled u.
The side opposite the right angle is known as the hypotenuse. The other sides of the
triangle are described by their position relative to the acute angle u. One side is
opposite u and one is adjacent to u.
The trigonometric functions have names that are words, rather than single
letters such as f, g, and h. For example, the sine of U is the length of the side
opposite u divided by the length of the hypotenuse:
sides from the point of view of an acute
angle u
sin u=
length of side opposite u
.
length of hypotenuse
Input is the measure
of an acute angle.
Output is the ratio of
the lengths of the sides.
The ratio of lengths depends on angle u and thus is a function of u. The expression sin u
really means sin1u2, where sine is the name of the function and u, the measure of an
acute angle, is the input.
Here are the names of the six trigonometric functions, along with their
abbreviations:
Name
Abbreviation
Name
Abbreviation
sine
sin
cosecant
csc
cosine
cos
secant
sec
tangent
tan
cotangent
cot
Now, let u be an acute angle in a right triangle, as shown in Figure 5.20. The
length of the side opposite u is a, the length of the side adjacent to u is b, and the
length of the hypotenuse is c.
Length of the
hypotenuse
c
B
a
Length of the
side opposite u
u
A
Figure 5.20
b
C
Length of the side adjacent to u
Right Triangle Definitions of Trigonometric Functions
See Figure 5.20. The six trigonometric functions of the acute angle U are defined as follows:
sin u =
length of side opposite angle u
a
=
c
length of hypotenuse
csc u =
length of hypotenuse
c
=
a
length of side opposite angle u
cos u =
length of side adjacent to angle u
b
=
c
length of hypotenuse
sec u =
length of hypotenuse
c
=
length of side adjacent to angle u
b
tan u =
length of side opposite angle u
a
=
length of side adjacent to angle u
b
cot u =
length of side adjacent to angle u
b
= .
a
length of side opposite angle u
Each of the trigonometric functions of the acute angle u is positive. Observe that the
ratios in the second column in the box are the reciprocals of the corresponding
ratios in the first column.
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Study Tip
The word
SOHCAHTOA 1pronounced: so-cah-tow-ah2
is a way to remember the right triangle definitions of the three basic trigonometric functions,
sine, cosine, and tangent.
S
OH
()*
opp
hyp
æ
C
AH
()*
adj
hyp
æ
Sine
T
æ
Cosine
OA
()*
opp
adj
Tangent
“Some Old Hog Came Around Here and Took Our Apples.”
Figure 5.21 shows four right triangles of varying sizes. In each of the triangles,
u is the same acute angle, measuring approximately 56.3°. All four of these similar
triangles have the same shape and the lengths of corresponding sides are in the same
ratio. In each triangle, the tangent function has the same value for the angle
u: tan u = 32 .
6
4.5
a3
1.5
u
u
Figure 5.21 A particular acute
angle always gives the same ratio of
opposite to adjacent sides.
b2
tan u a
b
4
3
2
tan u u
u
1
6
4
3
2
tan u 3
1.5
1
3
2
4.5
3
tan u 3 2
In general, the trigonometric function values of U depend only on the size
of angle U and not on the size of the triangle.
EXAMPLE 1
B
c
a5
u
A
b 12
C
Figure 5.22
Evaluating Trigonometric Functions
Find the value of each of the six trigonometric functions of u in Figure 5.22.
Solution We need to find the values of the six trigonometric functions of u.
However, we must know the lengths of all three sides of the triangle (a, b, and c) to
evaluate all six functions. The values of a and b are given. We can use the Pythagorean
Theorem, c2 = a2 + b2, to find c.
a=5
b = 12
c2=a2+b2=52+122=25+144=169
c = 2169 = 13
Study Tip
The function values in the second
column are reciprocals of those in
the first column. You can obtain each
of these values by exchanging the
numerator and denominator of
the corresponding ratio in the first
column.
Now that we know the lengths of the three sides of the triangle, we apply the
definitions of the six trigonometric functions of u. Referring to these lengths as
opposite, adjacent, and hypotenuse, we have
sin u =
opposite
5
=
hypotenuse
13
csc u =
hypotenuse
13
=
opposite
5
cos u =
adjacent
12
=
hypotenuse
13
sec u =
hypotenuse
13
=
adjacent
12
tan u =
opposite
5
=
adjacent
12
cot u =
adjacent
12
=
.
opposite
5
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500 Chapter 5 Trigonometric Functions
B
1
Check Point
Find the value of each of the six
trigonometric functions of u in the figure.
c
a⫽3
u
A
EXAMPLE 2
c=3
A
b⫽4
C
Evaluating Trigonometric Functions
B
Find the value of each of the six trigonometric functions of u in Figure 5.23.
a=1
Solution We begin by finding b.
u
C
b
a 2 + b2 = c 2
Figure 5.23
2
2
1 + b = 3
Use the Pythagorean Theorem.
2
Figure 5.23 shows that a = 1 and c = 3.
2
12 = 1 and 32 = 9.
1 + b = 9
b2 = 8
Subtract 1 from both sides.
b = 28 = 222
Take the principal square root and simplify:
28 = 24 # 2 = 24 22 = 222.
B
c=3
A
a=1
Now that we know the lengths of the three sides of the triangle, we apply the
definitions of the six trigonometric functions of u.
u
b = 2兹2
C
sin u =
opposite
1
=
hypotenuse
3
csc u =
cos u =
adjacent
222
=
hypotenuse
3
sec u =
tan u =
opposite
1
=
adjacent
222
Figure 5.23 (repeated,
showing b = 2 22)
hypotenuse
3
= = 3
opposite
1
hypotenuse
3
=
adjacent
222
adjacent
222
cot u =
=
= 222
opposite
1
Because fractional expressions are usually written without radicals in the denominators, we simplify the values of tan u and sec u by rationalizing the denominators:
tan u=
兹2
兹2
兹2
1
1
=
=
=
22
4
2兹2
2兹2 兹2
We are multiplying by 1 and
1
.
2兹2
兹2
3兹2
3兹2
3
3
=
=
=
.
22
4
2兹2
2兹2 兹2
We are multiplying by 1 and
3
.
2兹2
not changing the value of
Check Point
sec u=
not changing the value of
2
Find the value of each
of the six trigonometric functions of u in
the figure. Express each value in simplified form.
�
Find function values for
p
p
30°a b, 45°a b, and
6
4
p
60°a b.
3
B
c⫽5
A
u
b
a⫽1
C
Function Values for Some Special Angles
p
A 45°, or
radian, angle occurs frequently in trigonometry. How do we find the
4
values of the trigonometric functions of 45°? We construct a right triangle with a
45° angle, as shown in Figure 5.24. The triangle actually has two 45° angles. Thus, the
triangle is isosceles—that is, it has two sides of the same length. Assume that each
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Section 5.2 Right Triangle Trigonometry
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leg of the triangle has a length equal to 1. We can find the length of the hypotenuse
using the Pythagorean Theorem.
兹2
1length of hypotenuse22 = 12 + 12 = 2
length of hypotenuse = 22
1
45
With Figure 5.24, we can determine the trigonometric function values for 45°.
1
Figure 5.24 An
isosceles right triangle
Evaluating Trigonometric Functions of 45°
EXAMPLE 3
Use Figure 5.24 to find sin 45°, cos 45°, and tan 45°.
Solution We apply the definitions of these three trigonometric functions. Where
appropriate, we simplify by rationalizing denominators.
sin 45=
length of side opposite 45
1
1
兹2
兹2
=
=
=
length of hypotenuse
兹2
兹2 兹2
2
Rationalize denominators
cos 45=
length of side adjacent to 45
兹2
兹2
1
1
=
=
=
length of hypotenuse
兹2
兹2 兹2
2
tan 45=
length of side opposite 45
1
= =1
length of side adjacent to 45
1
Check Point
3
Use Figure 5.24 to find csc 45°, sec 45°, and cot 45°.
When you worked Check Point 3, did you actually use Figure 5.24 or did you
use reciprocals to find the values?
csc 45=兹2
Take the reciprocal
of sin 45° = 1 .
兹2
sec 45=兹2
Take the reciprocal
of cos 45° = 1 .
兹2
cot 45=1
Take the reciprocal
of tan 45° = 1 .
1
Notice that if you use reciprocals, you should take the reciprocal of a function value
before the denominator is rationalized. In this way, the reciprocal value will not
contain a radical in the denominator.
p
Two other angles that occur frequently in trigonometry are 30°, or radian,
6
p
and 60°, or radian, angles. We can find the values of the trigonometric functions of
3
30° and 60° by using a right triangle. To form this right triangle, draw an equilateral
triangle—that is, a triangle with all sides the same length. Assume that each side has a
length equal to 2. Now take half of the equilateral triangle. We obtain the right triangle
in Figure 5.25.This right triangle has a hypotenuse of length 2 and a leg of length 1.The
other leg has length a, which can be found using the Pythagorean Theorem.
30
a2 + 12 = 2 2
2
兹3
a2 + 1 = 4
a2 = 3
a = 23
60
1
Figure 5.25 30°–60°–90° triangle
With the right triangle in Figure 5.25, we can determine the trigonometric
functions for 30° and 60°.
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502 Chapter 5 Trigonometric Functions
Evaluating Trigonometric Functions of 30° and 60°
EXAMPLE 4
30
Use Figure 5.25 to find sin 60°, cos 60°, sin 30°, and cos 30°.
Solution We begin with 60°. Use the angle on the lower left in Figure 5.25.
2
兹3
sin 60° =
length of side opposite 60°
23
=
length of hypotenuse
2
cos 60° =
length of side adjacent to 60°
1
=
length of hypotenuse
2
60
1
Figure 5.25 (repeated)
To find sin 30° and cos 30°, use the angle on the upper right in Figure 5.25.
Check Point
sin 30° =
length of side opposite 30°
1
=
length of hypotenuse
2
cos 30° =
length of side adjacent to 30°
23
=
length of hypotenuse
2
4
Use Figure 5.25 to find tan 60° and tan 30°. If a radical appears
in a denominator, rationalize the denominator.
Because we will often use the function values of 30°, 45°, and 60°, you should
learn to construct the right triangles in Figure 5.24, shown on the previous page, and
Figure 5.25. With sufficient practice, you will memorize the values in Table 5.2.
Table 5.2 Trigonometric Functions of Special Angles
U
�
Recognize and use
fundamental identities.
30° P
6
P
4
45° 60° sin U
1
2
22
2
23
2
cos U
23
2
22
2
1
2
tan U
23
3
1
23
P
3
Fundamental Identities
Many relationships exist among the six trigonometric functions. These relationships
are described using trigonometric identities. For example, csc u is defined as the
reciprocal of sin u. This relationship can be expressed by the identity
csc u =
1
.
sin u
This identity is one of six reciprocal identities.
Reciprocal Identities
sin u =
1
csc u
csc u =
1
sin u
cos u =
1
sec u
sec u =
1
cos u
tan u =
1
cot u
cot u =
1
tan u
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Two other relationships that follow from the definitions of the trigonometric
functions are called the quotient identities.
Quotient Identities
tan u =
sin u
cos u
cot u =
cos u
sin u
If sin u and cos u are known, a quotient identity and three reciprocal identities
make it possible to find the value of each of the four remaining trigonometric
functions.
Using Quotient and Reciprocal Identities
EXAMPLE 5
2
221
, find the value of each of the four remaining
and cos u =
5
5
trigonometric functions.
Given sin u =
Solution We can find tan u by using the quotient identity that describes tan u as
the quotient of sin u and cos u.
2
sin u
2
2
兹21
2兹21
5
2
5
tan u=
=
=
= =
=
cos u
21
兹21
兹21
兹21
兹21
5
兹21
5
Rationalize the denominator.
We use the reciprocal identities to find the value of each of the remaining three
functions.
csc u =
sec u=
1
1
5
=
=
sin u
2
2
5
1
1
5
兹21
5兹21
5
=
=
=
=
cos u
21
兹21
兹21 兹21
兹21
5
Rationalize the denominator.
cot u =
1
=
tan u
1
2
=
221
221
2
We found tan u =
2
. We could use tan u =
221
but then we would have to rationalize the
denominator.
2221
,
21
2
25
, find the value of each of the
and cos u =
3
3
four remaining trigonometric functions.
Check Point
5
Given sin u =
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504 Chapter 5 Trigonometric Functions
Other relationships among trigonometric functions follow from the
Pythagorean Theorem. Using Figure 5.26, the Pythagorean Theorem states that
c
a
u
b
a 2 + b2 = c2.
To obtain ratios that correspond to trigonometric functions, divide both sides of this
equation by c2.
Figure 5.26
2
2
a
b
a2
b2
+
2
2 =1 or a c b + a c b =1
c
c
a
In Figure 5.26, sin u = c ,
so this is (sin u)2.
b
In Figure 5.26, cos u = c ,
so this is (cos u)2.
Based on the observations in the voice balloons, we see that
1sin u22 + 1cos u22 = 1.
We will use the notation sin2 u for 1sin u22 and cos2 u for 1cos u22. With this
notation, we can write the identity as
sin2 u + cos2 u = 1.
Two additional identities can be obtained from a2 + b2 = c2 by dividing both
sides by b2 and a2, respectively. The three identities are called the Pythagorean
identities.
Pythagorean Identities
sin2 u + cos2 u = 1
1 + tan2 u = sec2 u
1 + cot2 u = csc2 u
Using a Pythagorean Identity
EXAMPLE 6
Given that sin u = 35 and u is an acute angle, find the value of cos u using a
trigonometric identity.
Solution We can find the value of cos u by using the Pythagorean identity
sin2 u + cos2 u = 1.
3 2
a b + cos2 u = 1
5
9
+ cos2 u = 1
25
We are given that sin u =
3 3 2
32
9
Square : a b = 2 =
.
5 5
25
5
cos2 u = 1 cos2 u =
cos u =
Thus, cos u =
9
25
16
25
Subtract
Simplify: 1 -
16
A 25
=
9
25
9
16
=
=
.
25
25
25
25
4
Because u is an acute angle, cos u is positive.
5
6
Given that sin u =
cos u using a trigonometric identity.
Use equal cofunctions
of complements.
9
from both sides.
25
4
.
5
Check Point
�
3
.
5
1
2
and u is an acute angle, find the value of
Trigonometric Functions and Complements
p
Two positive angles are complements if their sum is 90° or . For example, angles of
70° and 20° are complements because 70° + 20° = 90°. 2
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Section 5.2 Right Triangle Trigonometry
c
u
b
Figure 5.27
90 − u
a
505
Another relationship among trigonometric functions is based on angles that
are complements. Refer to Figure 5.27. Because the sum of the angles of any triangle
is 180°, in a right triangle the sum of the acute angles is 90°. Thus, the acute angles
are complements. If the degree measure of one acute angle is u, then the degree
measure of the other acute angle is 190° - u2. This angle is shown on the upper right
in Figure 5.27.
Let’s use Figure 5.27 to compare sin u and cos190° - u2.
sin u =
cos190° - u2 =
length of side opposite u
a
=
c
length of hypotenuse
length of side adjacent to 190° - u2
length of hypotenuse
=
a
c
Thus, sin u = cos190° - u2. If two angles are complements, the sine of one equals
the cosine of the other. Because of this relationship, the sine and cosine are called
cofunctions of each other. The name cosine is a shortened form of the phrase
complement’s sine.
Any pair of trigonometric functions f and g for which
f1u2 = g190° - u2 and
g1u2 = f190° - u2
are called cofunctions. Using Figure 5.27, we can show that the tangent and
cotangent are also cofunctions of each other. So are the secant and cosecant.
Cofunction Identities
The value of a trigonometric function of u is equal to the cofunction of the
complement of u. Cofunctions of complementary angles are equal.
sin u = cos190° - u2
tan u = cot190° - u2
sec u = csc190° - u2
If u is in radians, replace 90° with
cos u = sin190° - u2
cot u = tan190° - u2
csc u = sec190° - u2
p
.
2
Using Cofunction Identities
EXAMPLE 7
Find a cofunction with the same value as the given expression:
p
a. sin 72°
b. csc .
3
Solution Because the value of a trigonometric function of u is equal to the
cofunction of the complement of u, we need to find the complement of each angle.
p
We do this by subtracting the angle’s measure from 90° or its radian equivalent, .
2
a. sin 72=cos(90-72)=cos 18
We have a function and
its cofunction.
b. csc
p
p
p
2p
p
3p
=sec a - b =sec a - b =sec
2
3
3
6
6
6
We have a cofunction
and its function.
Check Point
a. sin 46°
Perform the subtraction using the
least common denominator, 6.
7
Find a cofunction with the same value as the given expression:
b. cot
p
.
12
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506 Chapter 5 Trigonometric Functions
�
Evaluate trigonometric functions
with a calculator.
Using a Calculator to Evaluate Trigonometric Functions
The values of the trigonometric functions obtained with the special triangles are
exact values. For most acute angles other than 30°, 45°, and 60°, we approximate the
value of each of the trigonometric functions using a calculator. The first step is to set
the calculator to the correct mode, degrees or radians, depending on how the acute
angle is measured.
Most calculators have keys marked 冷SIN 冷, 冷COS 冷, and 冷TAN 冷. For example, to
find the value of sin 30°, set the calculator to the degree mode and enter 30 冷SIN 冷 on
most scientific calculators and 冷SIN 冷 30 冷ENTER 冷 on most graphing calculators.
Consult the manual for your calculator.
To evaluate the cosecant, secant, and cotangent functions, use the key for the
respective reciprocal function, 冷SIN 冷, 冷COS 冷, or 冷TAN 冷, and then use the reciprocal
key. The reciprocal key is 冷1>x 冷 on many scientific calculators and 冷x -1 冷 on many
p
graphing calculators. For example, we can evaluate sec
using the following
12
reciprocal relationship:
sec
p
=
12
1
cos
p
12
.
Using the radian mode, enter one of the following keystroke sequences:
Many Scientific Calculators
冷p 冷 冷 , 冷 12 冷 = 冷 冷COS 冷 冷1>x 冷
Many Graphing Calculators
冷1 冷 冷COS 冷 冷1 冷 冷p 冷 冷 , 冷 12 冷2 冷 冷2 冷 冷x -1 冷 冷ENTER 冷.
Rounding the display to four decimal places, we obtain sec
EXAMPLE 8
p
L 1.0353.
12
Evaluating Trigonometric Functions with a Calculator
Use a calculator to find the value to four decimal places:
a. cos 48.2°
b. cot 1.2.
Solution
Scientific Calculator Solution
Function
Mode
Keystrokes
Display, Rounded to
Four Decimal Places
a. cos 48.2°
Degree
48.2 冷COS 冷
0.6665
b. cot 1.2
Radian
1.2 冷TAN 冷 冷1>x 冷
0.3888
Graphing Calculator Solution
Function
Mode
Keystrokes
Display, Rounded to
Four Decimal Places
a. cos 48.2°
Degree
冷COS 冷 48.2 冷ENTER 冷
0.6665
b. cot 1.2
Radian
冷1 冷 冷TAN 冷 1.2 冷2 冷 冷x -1 冷 冷ENTER 冷
0.3888
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Section 5.2 Right Triangle Trigonometry
8
Check Point
a. sin 72.8°
�
Use right triangle trigonometry
to solve applied problems.
507
Use a calculator to find the value to four decimal places:
b. csc 1.5.
Applications
Many applications of right triangle trigonometry involve the angle made with an
imaginary horizontal line. As shown in Figure 5.28, an angle formed by a horizontal
line and the line of sight to an object that is above the horizontal line is called the
angle of elevation. The angle formed by a horizontal line and the line of sight to an
object that is below the horizontal line is called the angle of depression. Transits and
sextants are instruments used to measure such angles.
Observer
located
here
er
serv
ight
fs
ne o
Li
Lin
e of
ob
ove
ab
Angle of elevation
Horizontal
Angle of depression
sigh
t be
low
obs
erve
r
Figure 5.28
EXAMPLE 9
Problem Solving Using an Angle of Elevation
Sighting the top of a building, a surveyor measured the angle of elevation to be 22°.
The transit is 5 feet above the ground and 300 feet from the building. Find the
building’s height.
Solution The situation is illustrated in Figure 5.29. Let a be the height of the
portion of the building that lies above the transit. The height of the building is the
transit’s height, 5 feet, plus a. Thus, we need to identify a trigonometric function that
will make it possible to find a. In terms of the 22° angle, we are looking for the side
opposite the angle. The transit is 300 feet from the building, so the side adjacent to
the 22° angle is 300 feet. Because we have a known angle, an unknown opposite side,
and a known adjacent side, we select the tangent function.
a
Transit
h
t
Line
igh
of s
5 feet
22°
300 feet
Figure 5.29
a
tan 22=
300
Length of side opposite the 22° angle
Length of side adjacent to the 22° angle
a = 300 tan 22°
a L 121
Multiply both sides of the equation by 300.
Use a calculator in the degree mode.
The height of the part of the building above the transit is approximately 121 feet. Thus,
the height of the building is determined by adding the transit’s height, 5 feet, to 121 feet.
h L 5 + 121 = 126
The building’s height is approximately 126 feet.
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508 Chapter 5 Trigonometric Functions
Check Point
9
The irregular blue
shape in Figure 5.30 represents a
lake. The distance across the lake, a,
is unknown. To find this distance, a
surveyor took the measurements
shown in the figure. What is the
distance across the lake?
B
a
24°
Figure 5.30
A
750 yd
C
If two sides of a right triangle are known, an appropriate trigonometric function
can be used to find an acute angle u in the triangle.You will also need to use an inverse
trigonometric key on a calculator. These keys use a function value to display the acute
angle u. For example, suppose that sin u = 0.866. We can find u in the degree mode by
using the secondary inverse sine key, usually labeled 冷SIN -1 冷. The key 冷SIN -1 冷 is not a
button you will actually press. It is the secondary function for the button labeled 冷SIN 冷.
Many Scientific Calculators:
.866 2nd SIN
Pressing 2nd SIN
accesses the inverse
sine key, SIN−1 .
Many Graphing Calculators:
2nd SIN .866 ENTER
The display should show approximately 59.99, which can be rounded to 60. Thus, if
sin u = 0.866, then u L 60°.
EXAMPLE 10 Determining the Angle of Elevation
A building that is 21 meters tall casts a shadow 25 meters long. Find the angle of
elevation of the sun to the nearest degree.
Solution The situation is illustrated in Figure 5.31. We are asked to find u.
Angle of elevation
21 m
u
25 m
Figure 5.31
We begin with the tangent function.
side opposite u
21
=
tan u =
side adjacent to u
25
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Section 5.2 Right Triangle Trigonometry
509
We use a calculator in the degree mode to find u.
Many Scientific Calculators:
( 21÷25 ) 2nd TAN
Pressing 2nd TAN
accesses the inverse
tangent key, TAN−1 .
Many Graphing Calculators:
2nd TAN ( 21÷25 ) ENTER
The display should show approximately 40. Thus, the angle of elevation of the sun is
approximately 40°.
10
Check Point
A flagpole that is 14 meters tall casts a shadow 10 meters long.
Find the angle of elevation of the sun to the nearest degree.
The Mountain Man
In the 1930s, a National Geographic team headed by Brad Washburn
used trigonometry to create a map of the 5000-square-mile region of
the Yukon, near the Canadian border. The team started with aerial
photography. By drawing a network of angles on the photographs, the
approximate locations of the major mountains and their rough heights
were determined. The expedition then spent three months on foot to
find the exact heights. Team members established two base points a
known distance apart, one directly under the mountain’s peak. By
measuring the angle of elevation from one of the base points to the
peak, the tangent function was used to determine the peak’s height.
The Yukon expedition was a major advance in the way maps are made.
Exercise Set 5.2
Practice Exercises
5.
In Exercises 1–8, use the Pythagorean Theorem to find the length
of the missing side of each right triangle. Then find the value of
each of the six trigonometric functions of u.
1.
B
2.
B
26
10
u
A
C
B
6. B
6
C
41
u
9
A
u
A
u
7.
A
C
12
3. B
4. B
u
C
15
u
21
17
29
B
35
C
8
A
40
C
A
8.
C
24
u
C
21
A
B
u
25
A
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510 Chapter 5 Trigonometric Functions
In Exercises 9–16, use the given triangles to evaluate each
expression. If necessary, express the value without a square root in
the denominator by rationalizing the denominator.
In Exercises 39–48, use a calculator to find the value of the
trigonometric function to four decimal places.
39. sin 38°
40. cos 21°
41. tan 32.7°
42. tan 52.6°
43. csc 17°
44. sec 55°
30
2
1
60
45
9. cos 30°
10. tan 30°
11. sec 45°
p
13. tan
3
p
p
15. sin - cos
4
4
46. sin
3p
10
47. cot
p
12
48. cot
p
18
In Exercises 49–54, find the measure of the side of the right triangle
whose length is designated by a lowercase letter. Round answers to
the nearest whole number.
1
1
p
10
兹3
45
兹2
45. cos
49.
12. csc 45°
p
14. cot
3
p
p
16. tan + csc
4
6
B
50.
a
a
37
A
C
250 cm
In Exercises 17–20, u is an acute angle and sin u and cos u are
given. Use identities to find tan u, csc u, sec u, and cot u. Where
necessary, rationalize denominators.
15
17
17. sin u =
8
,
17
18. sin u =
3
,
5
cos u =
4
5
19. sin u =
1
,
3
cos u =
2 22
3
20. sin u =
6
,
7
cos u =
213
7
cos u =
61
A
51.
B
53.
21. sin u =
6
7
22. sin u =
7
8
23. sin u =
239
8
24. sin u =
221
5
p
p
+ cos2
9
9
29. sec2 23° - tan2 23°
28. sin2
p
p
+ cos2
10
10
34
32. sin 19°
33. csc 25°
34. csc 35°
2p
37. cos
5
A
C
C
B
c
16 m
23
A
C
54.
B
23 yd
44
A
b
C
30. csc2 63° - cot2 63°
31. sin 7°
p
9
a
26. cos 53° sec 53°
In Exercises 31–38, find a cofunction with the same value as the
given expression.
35. tan
13 m
b
In Exercises 25–30, use an identity to find the value of each
expression. Do not use a calculator.
27. sin2
C
B
34
A
10 cm
52.
220 in.
In Exercises 21–24, u is an acute angle and sin u is given. Use the
Pythagorean identity sin2 u + cos2 u = 1 to find cos u.
25. sin 37° csc 37°
B
36. tan
p
7
3p
38. cos
8
In Exercises 55–58, use a calculator to find the value of the acute
angle u to the nearest degree.
55. sin u = 0.2974
56. cos u = 0.8771
57. tan u = 4.6252
58. tan u = 26.0307
In Exercises 59–62, use a calculator to find the value of the acute
angle u in radians, rounded to three decimal places.
59. cos u = 0.4112
60. sin u = 0.9499
61. tan u = 0.4169
62. tan u = 0.5117
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Section 5.2 Right Triangle Trigonometry
Practice Plus
In Exercises 63–68, find the exact value of each expression. Do not
use a calculator.
tan
63.
2
p
3
1
-
2
2
p
cot
4
csc
2
65. 1 + sin 40° + sin 50°
75. A tower that is 125 feet tall casts a shadow 172 feet long. Find
the angle of elevation of the sun to the nearest degree.
2
1
64.
p
sec
6
511
p
6
125 ft
2
66. 1 - tan 10° + csc 80°
u
67. csc 37° sec 53° - tan 53° cot 37°
172 ft
68. cos 12° sin 78° + cos 78° sin 12°
In Exercises 69–70, express the exact value of each function as a
single fraction. Do not use a calculator.
76. The Washington Monument is 555 feet high. If you are standing one quarter of a mile, or 1320 feet, from the base of the
monument and looking to the top, find the angle of elevation
to the nearest degree.
Washington
Monument
p
69. If f1u2 = 2 cos u - cos 2u, find f a b.
6
u
p
70. If f1u2 = 2 sin u - sin , find f a b.
2
3
555 ft
u
1
p
71. If u is an acute angle and cot u = , find tan a - u b.
4
2
72. If u is an acute angle and cos u =
1
p
, find csc a - u b.
3
2
1320 ft
77. A plane rises from take-off and flies at an angle of 10° with
the horizontal runway. When it has gained 500 feet, find the
distance, to the nearest foot, the plane has flown.
Application Exercises
B
73. To find the distance across a lake, a surveyor took the measurements shown in the figure. Use these measurements to
determine how far it is across the lake. Round to the nearest
yard.
c=?
500 ft
10°
A
C
78. A road is inclined at an angle of 5°. After driving 5000 feet
along this road, find the driver’s increase in altitude. Round
to the nearest foot.
B
a=?
B
a=?
5000 ft
40°
A
5°
C
630 yd
74. At a certain time of day, the angle of elevation of the sun is 40°.
To the nearest foot, find the height of a tree whose shadow
is 35 feet long.
A
C
79. A telephone pole is 60 feet tall. A guy wire 75 feet long is
attached from the ground to the top of the pole. Find the
angle between the wire and the pole to the nearest degree.
u
60 ft
75 ft
h
40°
35 ft
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512 Chapter 5 Trigonometric Functions
80. A telephone pole is 55 feet tall. A guy wire 80 feet long is
attached from the ground to the top of the pole. Find the
angle between the wire and the pole to the nearest degree.
Writing in Mathematics
81. If you are given the lengths of the sides of a right triangle,
describe how to find the sine of either acute angle.
82. Describe one similarity and one difference between the
definitions of sin u and cos u, where u is an acute angle of a
right triangle.
83. Describe the triangle used to find the trigonometric
functions of 45°.
84. Describe the triangle used to find the trigonometric
functions of 30° and 60°.
85. What is a trigonometric identity?
86. Use words (not an equation) to describe one of the
reciprocal identities.
87. Use words (not an equation) to describe one of the quotient
identities.
88. Use words (not an equation) to describe one of the
Pythagorean identities.
89. Describe a relationship among trigonometric functions that
is based on angles that are complements.
93. Use a calculator in the radian mode to fill in the values in
cos u - 1
the following table. Then draw a conclusion about
u
as u approaches 0.
0.4
U
0.3
0.2
0.1
0.01 0.001 0.0001 0.00001
cos U
cos U ⴚ 1
U
Critical Thinking Exercises
Make Sense? In Exercises 94–97, determine whether each
statement makes sense or does not make sense, and explain
your reasoning.
94. For a given angle u, I found a slight increase in sin u as the
size of the triangle increased.
95. Although I can use an isosceles right triangle to determine
the exact value of sin p4 , I can also use my calculator to
obtain this value.
sin u
1
96. I can rewrite tan u as
, as well as
.
cot u
cos u
97. Standing under this arch, I can determine its height by
measuring the angle of elevation to the top of the arch and
my distance to a point directly under the arch.
90. Describe what is meant by an angle of elevation and an angle
of depression.
91. Stonehenge, the famous “stone circle” in England, was built
between 2750 B.C. and 1300 B.C. using solid stone blocks
weighing over 99,000 pounds each. It required 550 people to
pull a single stone up a ramp inclined at a 9° angle. Describe
how right triangle trigonometry can be used to determine
the distance the 550 workers had to drag a stone in order to
raise it to a height of 30 feet.
Delicate Arch in Arches National Park, Utah
Technology Exercises
92. Use a calculator in the radian mode to fill in the values in the
sin u
following table. Then draw a conclusion about
as u
u
approaches 0.
In Exercises 98–101, determine whether each statement is true or
false. If the statement is false, make the necessary change(s) to
produce a true statement.
98.
tan 45°
= tan 3°
tan 15°
100. sin 45° + cos 45° = 1
U
0.4
0.3
0.2
0.1
0.01
0.001
0.0001
0.00001
99. tan2 15° - sec2 15° = - 1
101. tan2 5° = tan 25°
sin U
102. Explain why the sine or cosine of an acute angle cannot be
greater than or equal to 1.
sin U
U
103. Describe what happens to the tangent of an acute angle as
the angle gets close to 90°. What happens at 90°?
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Section 5.3 Trigonometric Functions of Any Angle
104. From the top of a 250-foot lighthouse, a plane is sighted
overhead and a ship is observed directly below the plane.
The angle of elevation of the plane is 22° and the angle of
depression of the ship is 35°. Find a. the distance of the
ship from the lighthouse; b. the plane’s height above the
water. Round to the nearest foot.
Preview Exercises
Exercises 105–107 will help you prepare for the material covered
in the next section. Use these figures to solve Exercises 105–106.
y
105. a. Write a ratio that expresses sin u for the right triangle in
Figure (a).
b. Determine the ratio that you wrote in part (a) for
Figure (b) with x = - 3 and y = 4. Is this ratio positive
or negative?
106. a. Write a ratio that expresses cos u for the right triangle
in Figure (a).
b. Determine the ratio that you wrote in part (a) for
Figure (b) with x = - 3 and y = 5. Is this ratio positive
or negative?
107. Find the positive angle u¿ formed by the terminal side of u
and the x-axis.
a.
y
P ⫽ (x, y)
r
y
b.
u
u
x
y
x
x
5p
u⬘
Section
5.3
Objectives
� Use the definitions of
�
�
u⫽ 6
x
(b) u lies in
quadrant II.
(a) u lies in
quadrant I.
�
x
u⬘
y
x
y
u ⫽ 345⬚
P ⫽ (x, y)
r
trigonometric functions of
any angle.
Use the signs of the
trigonometric functions.
Find reference angles.
Use reference angles to
evaluate trigonometric
functions.
513
Trigonometric Functions of Any Angle
T
here is something comforting in the repetition of
some of nature’s patterns. The
ocean level at a beach varies
between high and low tide
approximately every 12 hours.
The number of hours of
daylight oscillates from a
maximum on the summer
solstice, June 21, to a minimum
on the winter solstice, December 21. Then it increases to the
same maximum the following
June 21. Some believe that
cycles, called biorhythms,
represent physical, emotional,
and intellectual aspects of our
lives. Throughout the remainder of this chapter, we will
see how the trigonometric
functions are used to model
phenomena that occur again and again. To do this, we need to move beyond right
triangles.