Section 5.2 Trigonometric Functions 5-1

Chapter 5 Trigonometric Functions

5.1 Angles

■ Basic Terminology ■ Degree Measure ■ Standard Position ■ Coterminal Angles

Key Terms: vertex of an angle, initial side, terminal side, positive angle, negative angle, quadrantal angle

Basic Terminology

A counterclockwise rotation generates a ____________ measure, and a clockwise rotation generates a

____________ measure.

Degree Measure

EXAMPLE 1 Finding the Complement and the Supplement of an Angle

For an angle measuring 40°, find the measure of (a) its complement and (b) its supplement.

EXAMPLE 2 Finding Measures of Complementary and Supplementary Angles

Find the measure of each marked angle in the figure.

5-2 Chapter 5 Trigonometric Functions

Standard Position

An angle is in ____________ _____________ if its vertex is at the origin and its initial side lies on the

positive x-axis. The figure shows ranges of angle measures for each quadrant when o o0 360 .

Quadrantal Angles

Angles in standard position whose terminal sides lie on the __________ or __________, such as angles

with measures o o o90 , 180 , 270 ,and so on, are quadrantal angles.

Coterminal Angles

Angles with measures o60 and o420 have the same initial side and the same terminal side, but different

amounts of rotation. Such angles are ____________ ____________. Their measures differ by a multiple

of ________.

EXAMPLE 5 Finding Measures of Coterminal Angles

Find the angles of least positive measure that are coterminal with each angle.

(a) o908 (b) o75 (c) o800

EXAMPLE 6 Analyzing the Revolutions of a CD Player

CD Players always spin at the same speed. Suppose a player makes 480 revolutions per min. Through

how many degrees will a point on the edge of a CD move in 2 sec?

Section 5.2 Trigonometric Functions 5-3

5.2 Trigonometric Functions

■ Trigonometric Functions ■ Quadrantal Angles ■ Reciprocal Identities

■ Signs and Ranges of Function Values ■ Pythagorean Identities ■ Quotient Identities

Key Terms: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), cosecant (csc)

Trigonometric Functions

Let ( , )x y be a point other than the origin on the terminal side of an angle in standard position. The

distance from the point to the origin is 2 2 .r x y The six trigonometric functions of are defined as

follows.

siny

r cos

x

r tan ( 0)

yx

x

csc ( 0)r

yy

sec ( 0)r

xx

cot ( 0)x

yy

Reciprocal Identities

For all angles for which both functions are defined, the following identities hold.

1

sincsc

1

cossec

1

tancot

1

cscsin

1

seccos

1

cottan

EXAMPLES Finding Function Values of an Angle

The terminal side of an angle in standard position passes through the point given. Find the values of

the six trigonometric functions of angle .

Example 1) through the point (8, 15). Example 2) through the point ( 3, 4).

5-4 Chapter 5 Trigonometric Functions

EXAMPLE 3 Finding Function Values of an Angle

Find the six trigonometric functions of the angle in standard position, if the terminal side of is

defined by 2 0, 0.x y x

EXAMPLE 4 Finding Function Values of Quadrantal Angles

Find the values of the six trigonometric functions for each angle.

(a) an angle of o90

(b) an angle in standard position with terminal

side through ( 3, 0)

Conditions for Undefined Function Values

Identify the terminal side of a quadrantal angle.

If the terminal side of the quadrantal angle lies along the y-axis, then the tangent and secant functions

are undefined.

If the terminal side of the quadrantal angle lies along the x-axis, then the cotangent and cosecant

functions are undefined.

The values given in this table can be found with a calculator that has trigonometric function keys. Make

sure the calculator is set in degree mode. One of the most common errors involving calculators in

trigonometry occurs when the calculator is set for radian measure, rather than degree measure.

Which row in the table gives the trigonometric function values for an angle measuring −270°?

Measuring 1350°?

Section 5.2 Trigonometric Functions 5-5

EXAMPLE 5 Using the Reciprocal Identities

Find each function value.

(a) cos , given that 5

sec3

(b) sin , given that 12

csc2

Signs of Function Values

θ in Quadrant sin θ cos θ tan θ cot θ sec θ csc θ

I

II

III

IV

EXAMPLE 6 Determining Signs of Functions of Nonquadrantal Angles

Determine the signs of the trigonometric functions of an angle in standard position with the given

measure.

(a) o87 (b) o300 (c) o200

EXAMPLE 7 Identifying the Quadrant of an Angle

Identify the quadrant(s) of an angle that satisfies the given conditions.

(a) sin 0, tan 0 (b) cos 0, sec 0

5-6 Chapter 5 Trigonometric Functions

Trigonometric

Function of

Range

(Interval Notation)

sin , cos [ 1, 1]

tan , cot ( , )

sec , csc ( , 1] [1, )

EXAMPLE 8 Deciding Whether a Value Is in the Range of a Trigonometric Function

Decide whether each statement is possible or impossible.

(a) sin 2.5 (b) tan 110.47 (c) sec 0.6

EXAMPLE 9 Finding All Function Values Given One Value and the Quadrant

Suppose that angle is in quadrant II and 2

sin .3

Find the values of the other five trigonometric

functions.

Pythagorean Identities

For all angles for which the function values are defined, the following identities hold.

2 2sin cos 1 2 2

tan 1 sec 2 21 cot csc

Quotient Identities

For all angles for which the denominators are not zero, the following identities hold.

sin

tancos

cos

cotsin

EXAMPLES: Find Function Values

Ex 10) Find sin and tan , given that

3cos

4 and sin 0.

Ex 11) Find sin and cos , given that

4tan

3 and is in quadrant III.

Section 5.3 Evaluating Trigonometric Functions 5-7

5.3 Evaluating Trigonometric Functions

■ Right-Triangle-Based Definitions of the Trigonometric Functions ■ Cofunctions ■ Trigonometric

Function Values of Special Angles ■ Reference Angles ■ Finding Function Values Using a

Calculator ■ Finding Angle Measures

Key Terms: side opposite, side adjacent, cofunctions, reference angle

Right-Triangle-Based Definitions of the Trigonometric Functions

In the figure, the side of length y is called the _____________

_____________ angle A, and the side of length x is called the

_____________ _____________ to angle A.

Right-Triangle-Based Definitions of Trigonometric Functions

Let A represent any acute angle in standard position.

sin csc

cos sec

tan cot

y rA A

r y

x rA A

r x

y xA A

x y

EXAMPLE 1 Finding Trigonometric Function Values of an Acute Angle

Find the sine, cosine, and tangent values for angles A and B in

the right triangle in the figure.

Cofunctions

________ ________

________ ________

________ ________

aA B

c

aA B

b

cA B

b

Cofunction Identities

For any acute angle A, cofunction values of complementary angles are equal.

sin cos 90 sec csc 90 tan cot 90

cos sin 90 csc sec 90 cot tan 90

A A A A A A

A A A A A A

5-8 Chapter 5 Trigonometric Functions

EXAMPLE 2 Writing Functions in Terms of Cofunctions

Write each function in terms of its cofunction.

(a) cos 52° (b) tan 71° (c) sec 24°

Trigonometric Function Values of Special Angles

EXAMPLE 3 Finding Trigonometric Function Values for 60°

Find the six trigonometric function values for a 60° angle.

Function Values of Special Angles

sin cos tan cot sec csc

30°

45°

60°

Reference Angles A ___________________

_______________ for an angle , written ,

is the positive acute angle made by the terminal

side of angle and the _____-axis.

The reference angle is always found with

reference to the ______-________.

EXAMPLE 4 Finding Reference Angles

Find the reference angle for each angle.

(a) 218°

(b) 1387°

Reference Angle for , where 0 360

___________ ___________

___________ ___________

Section 5.3 Evaluating Trigonometric Functions 5-9

Special Angles as Reference Angles

EXAMPLE 5 Finding Trigonometric Function Values of a Quadrant III Angle

Find the values of the six trigonometric

functions for 210°.

Finding Trigonometric Function Values for Any Nonquadrantal Angle

Step 1 If 360 , or if 0 , then find a coterminal angle by adding or subtracting 360° as many

times as needed to get an angle greater than 0° but less than 360°.

Step 2 Find the reference angle .

Step 3 Find the trigonometric function values for reference angle .

Step 4 Determine the correct signs for the values found in Step 3. (Use the table of signs in Section 5.2,

if necessary.) This gives the values of the trigonometric functions for angle .

EXAMPLE 6 Finding Trigonometric Function Values Using Reference Angles

Find the exact value of each expression.

(a) cos 240 (b) tan675

Finding Function Values Using a Calculator

When evaluating trigonometric functions of angles given in degrees, remember that the calculator

must be set in ____________ mode.

EXAMPLE 7 Finding Function Values With a Calculator Find the value of each expression rounded to the nearest thousandth.

(a) sin 49° (b) sec97.977

(c) 1

cot51.4283 (d) sin 246

__________

__________

__________

x

y

r

5-10 Chapter 5 Trigonometric Functions

Finding Angle Measures

If x is a number in the appropriate range, then 1 1sin , cos ,x x

or 1tan ,x

gives the measure of an

angle whose sine, cosine, or tangent, respectively, is x.

EXAMPLE 8 Using Inverse Trigonometric Functions to Find Angles

Use a calculator to find an angle in the interval 0 , 90 that satisfies each condition.

(a) sin 0.96770915 (b) sec 1.0545829

EXAMPLE 9 Finding Angle Measures Given an Interval and a Function Value

Find all values of , if is in the interval 0 , 360 and 2

cos .2

Section 5.4 Solving Right Triangles 5-11

5.4 Solving Right Triangles

■ Significant Digits ■ Solving Triangles ■ Angles of Elevation or Depression ■ Bearing

■ Further Applications

Key Terms: exact number, significant digits, angle of elevation, angle of depression, bearing

EXAMPLE 1 Solving a Right Triangle Given an Angle and a Side

Solve right triangle ABC, if A = 34° 30′ = 34.5° and c = 12.7 in.

To maintain accuracy, always use given information as much as possible, and do not round in

intermediate steps.

EXAMPLE 2 Solving a Right Triangle Given Two Sides

Solve right triangle ABC, if A = 29.43 cm and c = 53.58 cm.

Angles of Elevation or Depression

In applications of right triangles, the angle of _____________________ from point X to point Y (above X)

is the acute angle formed by ray XY and a horizontal ray with endpoint at X. See the figure part (a). The

angle of _____________________from point X to point Y (below X) is the acute angle formed by ray XY

and a horizontal ray with endpoint X. See the figure part (b).

(a) (b)

Both the angle of elevation and the angle of depression are measured between the line of sight and

a_______________________ ______________.

5-12 Chapter 5 Trigonometric Functions

Solving an Applied Trigonometry Problem

Step 1 ____________ ____________ ____________, and label it with the given information. Label

the quantity to be found with a variable.

Step 2 Use the sketch to write an ___________________ relating the given quantities to the variable.

Step 3 _____________________ ________ ________________, and check that your answer makes

sense.

EXAMPLE 3 Finding a Length Given the Angle of Elevation

Pat Porterfield knows that when she stands 123 ft from the base of a flagpole, the angle of elevation to the

top of the flagpole is 26° 45′ = 26.75°. If her eyes are 5.30 ft above the ground, find the height of the

flagpole.

Step 1

Step 2

Step 3

EXAMPLE 4 Finding an Angle of Depression

From the top of a 210-ft cliff, David observes a lighthouse that is 430 ft offshore. Find the angle of

depression from the top of the cliff to the base of the lighthouse.

Section 5.4 Solving Right Triangles 5-13

Bearing

Method 1 When a single angle is given, such as 164°, it is understood that the bearing is measured in a

clockwise direction from due north. Sample bearings using Method 1 are shown below.

EXAMPLE 5 Solving a Problem Involving Bearing (Method 1) Radar stations A and B are on an east-west line, 3.7 km apart. Station A detects a plane at C, on a bearing

of 61°. Station B simultaneously detects the same plane, on a bearing of 331°. Find the distance from A to

C.

Method 2 The second method for expressing bearing starts with a north-south line and uses an acute

angle to show the direction, either east or west, from this line. Sample bearings using Method

2 are shown below.

5-14 Chapter 5 Trigonometric Functions

EXAMPLE 6 Solving a Problem Involving Bearing (Method 2)

A ship leaves port and sails on a bearing of N 47° E for 3.5

hr. It then turns and sails on a bearing of S 43° E for 4.0 hr.

If the ship’s rate of speed is 22 knots (nautical miles per

hour), find the distance that the ship is from port.

Further Applications

EXAMPLE 7 Using Trigonometry to Measure a Distance

The subtense bar method is a method that surveyors use to determine a small distance d between two

points P and Q. The subtense bar with length b is centered at Q and situated perpendicular to the line of

sight between P and Q. Angle is measured, and then the distance d can be determined.

(a) Find d when 1 23 12 ≈ 1.3867° and b = 2.0000 cm.

(b) How much change would there be in the value of d if measured 1″ ≈ 0.0003° larger?

EXAMPLE 8 Solving a Problem Involving Angles of Elevation

Francisco needs to know the height of a tree. From a given point

on the ground, he finds that the angle of elevation to the top of

the tree is 36.7°. He then moves back 50 ft. From the second

point, the angle of elevation to the top of the tree is 22.2°. See

the figure. Find the height of the tree to the nearest foot.