SECTION 5.2 Right Triangle Trigonometry · Section 5.2 Right Triangle Trigonometry 533 Now let , u...

532 Chapter 5 Trigonometric Functions

M ountain climbers have forever been fascinated by reaching the top of Mount Everest, sometimes with tragic results. The mountain, on Asia’s Tibet–Nepal border, is Earth’s highest, peaking at an incredible 29,035 feet. The

heights of mountains can be found using trigonometry . The word trigonometrymeans “ 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.

The Six Trigonometric Functions We begin the study of trigonometry by defi ning 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 uand 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 udivided by the length of the hypotenuse:

length of side opposite ulength of hypotenuse

sin u= .

Output is the ratio ofthe lengths of the sides.

Input is the measureof an acute angle.

The ratio of lengths depends on angle u and thus is a function of u. The expression sin u really means sin(u), 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:

Right Triangle Trigonometry SECTION 5.2

Objectives � Use right triangles to

evaluate trigonometric functions.

� Find function values for

30�ap6b , 45�ap

4b , and

60�ap3b .

� 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.

� Use right triangles to evaluate trigonometric functions.

Side adjacent to u

Hypotenuse

u

Side opposite u

FIGURE 5.19 Naming a right triangle’s sides from the point of view of an acute angle u

Name Abbreviation Name Abbreviation

sine sin cosecant csc

cosine cos secant sec

tangent tan cotangent cot

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Section 5.2 Right Triangle Trigonometry 533

Now, let u be an acute angle in a right triangle, as shown in Figu re 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.

Right Triangle Defi nitions of Trigonometric Functions

See Figure 5.20 . The six trigonometric functions of the acute angle U are defi ned as follows:

sin u =length of side opposite angle u

length of hypotenuse=

ac csc u =

length of hypotenuse

length of side opposite angle u=

ca

cos u =length of side adjacent to angle u


bc sec u =

length of hypotenuse

length of side adjacent to angle u=

cb

tan u =length of side opposite angle u

length of side adjacent to angle u=

ab cot u =

length of side adjacent to angle u

length of side opposite angle u=

ba

.

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 fi rst column.

GREAT QUESTION! Is there a way to help me remember the right triangle defi nitions of any of the trigonometric functions?

The word SOHCAHTOA (pronounced: so@cah@tow@ah)

may be helpful in remembering the defi nitions for sine, cosine, and tangent.

S O H C A H T O A

æ æ æSine Cosine Tangent

()* ()* ()*

opphyp

adjhyp

oppadj

“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 = 3

2.

b � 2

a � 3

u

4

6

u

1

1.5u

3

4.5

u

tan u � �32

ab tan u � �

32

64 tan u � �

32

1.51 tan u � �

32

4.53

FIGURE 5.21 A particular acute angle always gives the same ratio of opposite to adjacent sides.

A

B

C

a

b

c

Length of the side adjacent to u

Length of thehypotenuse

u

Length of theside opposite u

FIGURE 5.20

In general, the trigonometric function values of U depend only on the size of angle U and not on the size of the triangle.

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EXAMPLE 1 Evaluating Trigonometric Functions

Find the value of each of the six trigonometric functions of u in Figure 5.22 .

SOLUTION We need to fi nd 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 fi nd c.

c2=a2+b2=52+122=25+144=169

a = 5 b = 12

c = 2169 = 13

Now that we know the lengths of the three sides of the triangle, we apply the defi nitions of the six trigonometric functions of u. Referring to these lengths as opposite, adjacent, and hypotenuse, we have

sin u =opposite

hypotenuse=

513 csc u =

hypotenuse

opposite=

135

cos u =adjacent

hypotenuse=

1213 sec u =

hypotenuse

adjacent=

1312

tan u =opposite

adjacent=

512 cot u =

adjacent

opposite=

125

.

Check Point 1 Find the value of each of the six trigonometric functions of u in the fi gure.

EXAMPLE 2 Evaluating Trigonometric Functions

Find the value of each of the six trigonometric functions of u in Figure 5.23 .

SOLUTION We begin by fi nding b.

a2 + b2 = c2 Use the Pythagorean Theorem.

12 + b2 = 32 Figure 5.23 shows that a = 1 and c = 3.

1 + b2 = 9 12 = 1 and 32 = 9.

b2 = 8 Subtract 1 from both sides.

b = 28 = 222 Take the principal square root and simplify: 28 = 24 # 2 = 2422 = 222.

b � 12

a � 5c

u

B

CA

FIGURE 5.22

GREAT QUESTION! Do I have to use the defi nitions of the trigonometric functions to get the function values shown in the second column?

No. The function values in the second column are reciprocals of those in the fi rst column. You can obtain each of these values by interchanging the numerator and denominator of the corresponding ratio in the fi rst column.

● ● ●

b � 4

a � 3c

u

B

CA

a = 1c = 3

u

B

CAb

FIGURE 5.23

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Now that we know the lengths of the three sides of the triangle, we apply the defi nitions of the six trigonometric functions of u.

sin u =opposite

hypotenuse=

13 csc u =

hypotenuse

opposite=

31= 3

cos u =adjacent

hypotenuse=

2223 sec u =

hypotenuse

adjacent=

3

222

tan u =opposite

adjacent=

1

222 cot u =

adjacent

opposite=

2221

= 222

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:

a = 1c = 3

u

B

CAb = 2�2

FIGURE 5.23 (repeated, showing b = 222)

We are multiplying by 1 and

not changing the value of .

tan u= �1

222

1

222= = =

22

22

22

2 � 2

22

4

12�2

We are multiplying by 1 and

not changing the value of .

sec u= �3

222

3

222= = .=

22

22

322

2 � 2

322

4

32�2

Check Point 2 Find the value of each of the six trigonometric functions of u in the fi gure. Express each value in simplifi ed form.

Function Values for Some Special Angles

A 45�, or p

4 radian, angle occurs frequently in trigonometry. How do we fi nd the

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 leg of the triangle has a length equal to 1. We can fi nd the length of the hypotenuse using the Pythagorean Theorem.

(length of hypotenuse)2 = 12 + 12 = 2

length of hypotenuse = 22

With Figure 5.24 , we can determine the trigonometric function values for 45�.

EXAMPLE 3 Evaluating Trigonometric Functions of 45�

Use Figure 5.24 to fi nd sin 45�, cos 45�, and tan 45�.

SOLUTION We apply the defi nitions of these three trigonometric functions. Where appropriate, we simplify by rationalizing denominators.

b

a � 1c � 5

u

B

CA

� Find function values for

30�ap6b , 45�ap

4b , and

60�ap3b .

1

1�2

45�

FIGURE 5.24 An isosceles right triangle

● ● ●

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length of side opposite 45�length of hypotenuse

sin 45�= = �

Rationalize denominators.

1

22= =

1

22

1

22

1

22

22

22

22

22

22

2

22

2

length of side adjacent to 45�length of hypotenuse

cos 45�= = �= =

length of side opposite 45�length of side adjacent to 45�

11

tan 45�= = =1

Check Point 3 Use Figure 5.24 to fi nd 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 fi nd the values?

Take the reciprocal

csc 45�=22

of sin 45° = .1�2

Take the reciprocal

sec 45�=22

of cos 45° = .1�2

Take the reciprocal

cot 45�=1

of tan 45° = .11

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.

Two other angles that occur frequently in trigonometry are 30�, or p

6 radian, and

60�, or p

3 radian, angles. We can fi nd the values of the trigonometric functions of

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.

a2 + 12 = 22

a2 + 1 = 4

a2 = 3

a = 23

With the right triangle in Figure 5.25 , we can determine the trigonometric functions for 30� and 60�.

EXAMPLE 4 Evaluating Trigonometric Functions of 30� and 60�

Use Figure 5.25 to fi nd 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 .

sin 60� =length of side opposite 60�

length of hypotenuse=232

cos 60� =length of side adjacent to 60�


12

GREAT QUESTION! Can I use my calculator to evaluate trigonometric functions and skip this part of the section?

No. Later in the section you’ll learn that calculators provide approximate values of the trigonometric functions for most angles. The function values obtained in this part of the section are exact. When these exact values are irrational numbers, they cannot be obtained with a calculator.

1

2

�3

60�

30�

FIGURE 5.25 30�–60�–90� triangle

● ● ●

1

1�2

45�

FIGURE 5.24 An isosceles right triangle (repeated)

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If sin u and cos u are known, a quotient identity and three reciprocal identities make it possible to fi nd the value of each of the four remaining trigonometric functions.

To fi nd sin 30� and cos 30�, use the angle on the upper right in Fi gure 5.25 .

sin 30� =length of side opposite 30�


12

cos 30� =length of side adjacent to 30�

length of hypotenuse=232

● ● ●

Check Point 4 Use Figure 5.25 to fi nd 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 and Figure 5.25 , shown on the previous page. With suffi cient practice, you will memorize the values in Table 5.2 .

Table 5.2 Trigonometric Functions of Special Angles

U 30� �P

6 45� �

P

4 60� �

P

3

sin U 12

222

232

cos U 232

222

12

tan U 233

1 23

� Recognize and use fundamental identities.

Fundamental Identities Many relationships exist among the six trigonometric functions. These relationships are described using trigonometric identities . For example, csc u is defi ned 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 =

1sin u

cos u =1

sec u sec u =

1cos u

tan u =1

cot u cot u =

1tan u

Two other relationships that follow from the defi nitions of the trigonometric functions are called the quotient identities .

Quotient Identities

tan u =sin ucos u

cot u =cos usin u

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EXAMPLE 5 Using Quotient and Reciprocal Identities

Given sin u =25

and cos u =221

5, fi nd the value of each of the four remaining

trigonometric functions.

SOLUTION We can fi nd tan u by using the quotient identity that describes tan u as the quotient of sin u and cos u.

tan u= � �5

221= = ==

2

221=

2

221

221

221

2221

21221

5

sin ucos u

2

5 2

5

Rationalize the denominator.

We use the reciprocal identities to fi nd the value of each of the remaining three functions.

csc u =1

sin u=

125

=52

sec u= �= = ==1

cos u1

Rationalize the denominator.

5

221

5

221

221

221

5221

21221

5

cot u =1

tan u=

12

221

=221

2 We found tan u =

2

221. We could use tan u =

222121

, but then we would have to rationalize the

denominator. ● ● ●

Check Point 5 Given sin u =23

and cos u =253

, fi nd the value of each of the

four remaining trigonometric functions.

Other relationships among trigonometric functions follow from the Pythagorean Theorem. Using Figure 5.26 , the Pythagorean Theorem states that

a2 + b2 = c2.

To obtain ratios that correspond to trigonometric functions, divide both sides of this equation by c2.

In Figure 5.26, sin u = so this is (sin u)2.

,

ac

b2

c2

a2

c2a b2 b

ca b2=1 or =1+ +

ac In Figure 5.26, cos u =

so this is (cos u)2.,b

c

Based on the observations in the voice balloons, we see that

(sin u)2 + (cos u)2 = 1.

b

ac

u

FIGURE 5.26

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We will use the notation sin2 u for (sin u)2 and cos2 u for (cos u)2. 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

EXAMPLE 6 Using a Pythagorean Identity

Given that sin u = 35 and u is an acute angle, fi nd the value of cos u using a

trigonometric identity.

SOLUTION We can fi nd the value of cos u by using the Pythagorean identity

sin2 u + cos2 u = 1.

a 35b2

+ cos2 u = 1 We are given that sin u =35.

925

+ cos2 u = 1 Square 35

: a35b2

=32

52 =9

25.

cos2 u = 1 -925

Subtract 9

25 from both sides.

cos2 u =1625

Simplify: 1 -9

25=

2525

-9

25=

1625

.

cos u = A1625

=45

Because u is an acute angle, cos u is positive.

Thus, cos u =45

. ● ● ●

Check Point 6 Given that sin u = 12 and u is an acute angle, fi nd the value of

cos u using a trigonometric identity.

Trigonometric Functions and Complements

Two positive angles are complements if their sum is 90� or p

2. For example, angles of

70� and 20� are complements because 70� + 20� = 90�. 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 (90� - u). This angle is shown on the upper right in Figure 5.27 .

Let’s use Figure 5.27 to compare sin u and cos(90� - u).

sin u =length of side opposite u


ac

cos(90� - u) =length of side adjacent to (90� - u)


ac

Thus, sin u = cos(90� - u).

� Use equal cofunctions of complements.

b

90� − ua

c

u

FIGURE 5.27

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Because sin u = cos(90� - u), 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

f(u) = g(90� - u) and g(u) = f(90� - u)

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 = cos(90� - u) cos u = sin(90� - u)

tan u = cot(90� - u) cot u = tan(90� - u)

sec u = csc(90� - u) csc u = sec(90� - u)

If u is in radians, replace 90� with p

2.

EXAMPLE 7 Using Cofunction Identities

Find a cofunction with the same value as the given expression:

a. sin 72� b. csc p

3.

SOLUTION Because the value of a trigonometric function of u is equal to the cofunction of the complement of u, we need to fi nd the complement of each angle. We do this by subtracting the angle’s measure from 90� or its radian equivalent,

p

2.

a. sin 72�=cos(90�-72�)=cos 18�

We have a function andits cofunction.

b. -=seccscp

2

p

3

p

6

p

3a b -=sec =sec

3p

6

2p

6a b

We have a cofunctionand its function.

Perform the subtraction using theleast common denominator, 6.

Check Point 7 Find a cofunction with the same value as the given expression:

a. sin 46� b. cot p

12.

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 fi rst step is to set the calculator to the correct mode , degrees or radians, depending on how the acute angle is measured.

● ● ●

� Evaluate trigonometric functions with a calculator.

b

90� − ua

c

u

FIGURE 5.27 (repeated)

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Most calculators have keys marked � SIN � , � COS � , and � TAN � . For example, to fi nd the value of sin 30�, set the calculator to the degree mode and enter 30 � SIN � on most scientifi c calculators and � SIN � 30 � ENTER � on most graphing calculators. Consult the manual for your calculator.

Scientifi c Calculator Solution

Function Mode Keystrokes Display, Rounded to Four Decimal Places

a. cos 48.2� Degree 48.2 � COS � 0.6665

b. cot 1.2 Radian 1.2 � TAN � � 1>x � 0.3888

Graphing Calculator Solution

Function Mode Keystrokes Display, Rounded to Four Decimal Places

a. cos 48.2� Degree � COS �48.2 � ENTER � 0.6665

b. cot 1.2 Radian � ( � � TAN �1.2 � ) � � x-1 � � ENTER � 0.3888

Check Point 8 Use a calculator to fi nd the value to four decimal places:

a. sin 72.8� b. csc 1.5.

● ● ●

To evaluate the cosecant, secant, and cotangent functions, use the key for the

respective reciprocal function, � SIN � , � COS � , or � TAN � , and then use the reciprocal

key. The reciprocal key is � 1>x � on many scientifi c calculators and � x-1 � on many

graphing calculators. For example, we can evaluate sec p

12 using the following

reciprocal relationship:

sec p

12=

1

cos p

12

.

Using the radian mode, enter one of the following keystroke sequences:

Many Scientifi c Calculators

� p � � , �12 � = � � COS � � 1>x �

Many Graphing Calculators

� ( � � COS � � ( � � p � � , �12 � ) � � ) � � x-1 � � ENTER � .

Rounding the display to four decimal places, we obtain sec p

12� 1.0353.

EXAMPLE 8 Evaluating Trigonometric Functions with a Calculator

Use a calculator to fi nd the value to four decimal places:

a. cos 48.2� b. cot 1.2.

SOLUTION

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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.

� Use right triangle trigonometry to solve applied problems.

Line of sight above observer

Angle of elevation

Angle of depressionLine of sight below observer

Observerlocatedhere

Horizontal

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 fi nd 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.

ha

22°300 feet

5 feetLine of sightTransit

FIGURE 5.29

a

300tan 22�=

Length of side opposite the 22° angle

Length of side adjacent to the 22° angle

a = 300 tan 22� Multiply both sides of the equation by 300.

a � 121 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 � 5 + 121 = 126

The building’s height is approximately 126 feet. ● ● ●

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Check Point 9 The irregular blue shape in Figure 5.30 represents a lake. The distance across the lake, a, is unknown. To fi nd this distance, a surveyor took the measurements shown in the fi gure. What is the distance across the lake?

If two sides of a right triangle are known, an appropriate trigonometric function can be used to fi nd 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 fi nd u in the degree mode by using the secondary inverse sine key, usually labelled � SIN-1 � . The key � SIN-1 � is not a button you will actually press. It is the secondary function for the button labeled � SIN � .

.866 2nd SIN

Many Scientific Calculators:

Pressing 2nd SINaccesses the inversesine key, SIN−1 .

2nd SIN .866 ENTER

Many Graphing Calculators:

The display should show approximately 59.997, which can be rounded to 60. Thus, if sin u = 0.866, then u � 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 fi nd u.

750 ydA C

B

a

24°

FIGURE 5.30

25 m

21 m

u

Angle of elevation

FIGURE 5.31

We begin with the tangent function.

tan u =side opposite u

side adjacent to u=

2125

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We use tan u = 2125 and a calculator in the degree mode to fi nd u.

2521÷ TAN2nd)(

Many Scientific Calculators:

Pressing 2nd TANaccesses the inversetangent key, TAN−1 .

2nd TAN ENTER

Many Graphing Calculators:

2521÷ )(

The display should show approximately 40. Thus, the angle of elevation of the sun is approximately 40�. ● ● ●

Check Point 10 A fl agpole that is 14 meters tall casts a shadow 10 meters long. Find the angle of elevation of the sun to the nearest degree.

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 fi nd 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.

Blitzer Bonus ❘ ❘ The Mountain Man

1. Using lengths a, b, and c in the right triangle shown, the trigonometric functions of u are defi ned as follows:

b

ca

u

B

CA

Fill in each blank so that the resulting statement is true.

CONCEPT AND VOCABULARY CHECK

sin u = csc u =

cos u = sec u =

tan u = cot u = .

2. Using the defi nitions in Exercise 1, we refer to a as the length of the side angle u, b as the length of the side angle u, and c as the length of the .

3. True or false: The trigonometric functions of u in Exercise 1 depend only on the size of u and not on the size of the triangle.

4. According to the reciprocal identities,

1

csc u= ,

1sec u

= , and 1

cot u= .

5. According to the quotient identities,

sin ucos u

= and cos usin u

= .

6. According to the Pythagorean identities,

sin2 u + cos2 u = , 1 + tan2 u = ,

and 1 + cot2 u = .

7. According to the cofunction identities,

cos(90� - u) = , cot(90� - u) = ,

and csc(90� - u) = .

25 m

21 m

u

Angle ofelevation

FIGURE 5.31 (repeated)

M10_BLIT7240_05_SE_05.indd 544 13/10/12 10:23 AM


9. cos 30� 10. tan 30� 11. sec 45� 12. csc 45�

13. tan p

3 14. cot

p

3

15. sin p

4- cos

p

4 16. tan

p

4+ csc

p

6

In Exercises 17–20, u is an acute angle and sin u and cos u are given. Use identities to fi nd tan u, csc u, sec u, and cot u. Where necessary, rationalize denominators.

17. sin u =8

17, cos u =

1517

18. sin u =35

, cos u =45

19. sin u =13

, cos u =222

3

20. sin u =67

, cos u =213

7

In Exercises 21–24, u is an acute angle and sin u is given. Use the Pythagorean identity sin2 u + cos2 u = 1 to fi nd cos u.

21. sin u =67

22. sin u =78

23. sin u =239

8 24. sin u =

2215

In Exercises 25–30, use an identity to fi nd the value of each expression. Do not use a calculator.

25. sin 37� csc 37� 26. cos 53� sec 53�

27. sin2 p

9+ cos2

p

9 28. sin2

p

10+ cos2

p

10

29. sec2 23� - tan2 23� 30. csc2 63� - cot2 63�

In Exercises 31–38, fi nd a cofunction with the same value as the given expression.

31. sin 7� 32. sin 19� 33. csc 25� 34. csc 35�

35. tan p

9 36. tan

p

7

37. cos 2p5

38. cos 3p8

In Exercises 39–48, use a calculator to fi nd 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�

45. cos p

10 46. sin

3p10

47. cot p

12 48. cot

p

18

Practice Exercises In Exercises 1–8, use the Pythagorean Theorem to fi nd the length of the missing side of each right triangle. Then fi nd the value of each of the six trigonometric functions of u.

EXERCISE SET 5.2

1.

12

9

u

B

CA

2.

8

6

u

B

CA

3.

21

29

u

B

C A

4.

15

17

u

B

C A

5.

1026

u

B

C A

6.

40

41

u

B

C A

7.

21

35

u

B

C

A

8.

24

25

uB

C

A

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.

1

1�2

45�

45�

1

2�3

60�

30�

M10_BLIT7240_05_SE_05.indd 545 13/10/12 10:23 AM


In Exercises 49–54, fi nd the measure of the side of the right triangle whose length is designated by a lowercase letter. Round answers to the nearest whole number.

In Exercises 69–70, express the exact value of each function as a single fraction. Do not use a calculator.

69. If f(u) = 2 cos u - cos 2u, fi nd fap6b .

70. If f(u) = 2 sin u - sin u

2, fi nd fap

3b .

71. If u is an acute angle and cot u =14

, fi nd tanap2

- ub .

72. If u is an acute angle and cos u =13

, fi nd cscap2

- ub .

Application Exercises 73. To fi nd the distance across a lake, a surveyor took the

measurements shown in the fi gure. Use these measurements to determine how far it is across the lake. Round to the nearest yard.

630 ydA C

B

a = ?

40°

74. At a certain time of day, the angle of elevation of the sun is 40�. To the nearest foot, fi nd the height of a tree whose shadow is 35 feet long.

35 ft

h

40°

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.

172 ft

125 ft

u

49.

250 cm

a

B

CA37�

50.

10 cm

a

B

CA61�

51.

b

B

CA34�

220 in.

52.

a

B

CA34�

13 m

53.

16 mc

B

C A23�

54.

b

B

CA44�

23 yd

In Exercises 55–58, use a calculator to fi nd 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 fi nd 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

Practice Plus In Exercises 63–68, fi nd the exact value of each expression. Do not use a calculator.

63. tan p

32

-1

sec p

6

64. 1

cot p

4

-2

csc p

6

65. 1 + sin2 40� + sin2 50� 66. 1 - tan2 10� + csc2 80� 67. csc 37� sec 53� - tan 53� cot 37� 68. cos 12� sin 78� + cos 78� sin 12�

M10_BLIT7240_05_SE_05.indd 546 13/10/12 10:23 AM


83. Describe the triangle used to fi nd the trigonometric functions of 45�.

84. Describe the triangle used to fi nd 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. 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.

Technology Exercises 92. Use a calculator in the radian mode to fi ll in the values in

the following table. Then draw a conclusion about sin uu

as u

approaches 0.

U 0.4 0.3 0.2 0.1 0.01 0.001 0.0001 0.00001

sin U

sin UU

93. Use a calculator in the radian mode to fi ll in the values in the

following table. Then draw a conclusion about cos u - 1u

as u

approaches 0.

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, fi nd the angle of elevation to the nearest degree.

1320 ft

WashingtonMonument

555 ft

u

77. A plane rises from take-off and fl ies at an angle of 10� with the horizontal runway. When it has gained 500 feet, fi nd the distance, to the nearest foot, the plane has fl own.

500 ft10°

c = ?

A C

B

78. A road is inclined at an angle of 5�. After driving 5000 feet along this road, fi nd the driver’s increase in altitude. Round to the nearest foot.

5000 ft

C

B

A

a = ?5°

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.

60 ft75 ft

u

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 fi nd the sine of either acute angle. 82. Describe one similarity and one difference between the

defi nitions of sin u and cos u, where u is an acute angle of a right triangle.

U 0.4 0.3 0.2 0.1 0.01 0.001 0.0001 0.00001

cos U

cos U � 1U

M10_BLIT7240_05_SE_05.indd 547 13/10/12 10:23 AM


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.

96. I can rewrite tan u as 1

cot u, as well as

sin ucos 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.

Delicate Arch in Arches National Park, Utah

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 15�= tan 3� 99. tan2 15� - sec2 15� = -1

100. sin 45� + cos 45� = 1 101. tan2 5� = tan 25� 102. Explain why the sine or cosine of an acute angle cannot be

greater than or equal to 1. 103. Describe what happens to the tangent of an acute angle as

the angle gets close to 90�. What happens at 90�? 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 fi gures to solve Exercises 105–106.

y

x

r

u

y

x

P � (x, y)

(a) u lies inquadrant I.

y

x

r

u

y

x

P � (x, y)

(b) u lies inquadrant II.

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.

u � 345�

u�

y

x

b. y

xu�

u � 5p6

M10_BLIT7240_05_SE_05.indd 548 13/10/12 10:23 AM

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