Chapter 5 Trigonometric Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 5.1 Angles and Radian Measure

Chapter 5Trigonometric Functions

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

5.1 Angles and Radian Measure


Objectives:• Recognize and use the vocabulary of angles.• Use degree measure.• Use radian measure.• Convert between degrees and radians.• Draw angles in standard position.• Find coterminal angles.• Find the length of a circular arc.• Use linear and angular speed to describe motion on a circular path.


Angles

An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other the terminal side.


Angles (continued)

An angle is in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side lies along the positive x-axis.


Angles (continued)When we see an initial side and a terminal side in place, there are two kinds of rotations that could have generated the angle.Positive angles are generated by counterclockwise rotation. Thus, angle is positive. Negative angles are generated by clockwise rotation. Thus, angle is negative.


Angles (continued)

An angle is called a quadrantal angle if its terminal side lies on the x-axis or on the y-axis. Angle is an example of a quadrantal angle.


Measuring Angles Using Degrees

Angles are measured by determining the amount of rotation from the initial side to the terminal side. A complete rotation of the circle is 360 degrees, or 360°. An acute angle measures less than 90°.A right angle measures 90°.An obtuse angle measures more than 90° but less than

180°.A straight angle measures 180°.


Measuring Angles Using Radians

An angle whose vertex is at the center of the circle is called a central angle. The radian measure of any central angle of a circle is the length of the intercepted arc divided by the circle’s radius.


Definition of a Radian

One radian is the measure of the central angle of a circle that intercepts an arc equal in length to the radius of the circle.


Radian Measure


Example: Computing Radian Measure

A central angle, in a circle of radius 12 feet intercepts an arc of length 42 feet. What is the radian measure of ?

The radian measure of is 3.5 radians.

sr

42 feet12 feet

3.5


Conversion between Degrees and Radians


Example: Converting from Degrees to Radians

Convert each angle in degrees to radians:

a. 60°

b. 270°

c. –300°

radians60180

60 radians180

radians3

radians270180

270 radians180

3 radians2

radians300180

300 radians180

5 radians3


Example: Converting from Radians to Degrees

Convert each angle in radians to degrees:

a.

b.

c.

radians4

4 radians3

6 radians

radians 1804 radians

180

4 45

4 radians 1803 radians

4 1803

240 1806 radians radians

6 180

343.8


Drawing Angles in Standard PositionThe figure illustrates that when the terminal side makes

one full revolution, it forms an angle whose radian measure is The figure shows the quadrantal angles formed by 3/4, 1/2, and 1/4 of a revolution.

2 .


Example: Drawing Angles in Standard Position

Draw and label the angle in standard position:

x

y

4

Initial side

Terminal side

Vertex

The angle is negative. It is obtained by rotating the terminal side clockwise.

1 24 8

We rotate the terminal side

clockwise of a revolution.18



Draw and label the angle in standard position: 34

x

yThe angle is positive. It is obtained by rotating the terminal side counterclockwise.

3 3 24 8


counter clockwise of a revolution.38

Vertex

Terminal side

Initial side




x

y The angle is negative. It is obtained by rotating the terminal side clockwise.


clockwise of a revolution.78

7 7 24 8

Terminal side

Initial sideVertex




The angle is positive. It is obtained by rotating the terminal side counterclockwise.


counter clockwise of a revolution.138

13 13 24 8

x

y

Terminal side

Vertex

Initial side


Degree and Radian Measures of Angles Commonly Seen in Trigonometry

In the figure below, each angle is in standard position, so that the initial side lies along the positive x-axis.


Positive Angles in Terms of Revolutions of the Angle’s Terminal Side Around the Origin


Positive Angles in Terms of Revolutions of the Angle’s Terminal Side Around the Origin (continued)


Coterminal Angles

Two angles with the same initial and terminal sides butpossibly different rotations are called coterminal angles.


Example: Finding Coterminal Angles

Assume the following angles are in standard position. Find a positive angle less than 360° that is coterminal with each of the following:a. a 400° angle

400° – 360° = 40°b. a –135° angle

–135° + 360° = 225°


Example: Finding Coterminal Angles

Assume the following angles are in standard position. Find a positive angle less than that is coterminal with each of the following:

a. a angle

b. a angle

135 13 13 10 32

5 5 5 5

2

15 30 292

15 15 15 15


The Length of a Circular Arc


Example: Finding the Length of a Circular ArcA circle has a radius of 6 inches. Find the length of the

arc intercepted by a central angle of 45°. Express arc length in terms of Then round your answer to two decimal places.

We first convert 45° to radians:

.

radians 4545 45 radians180 180 4

s r (6 inches)4

6 inches 4 4.71 inches.


Definitions of Linear and Angular Speed


Linear Speed in Terms of Angular Speed


Example: Finding Linear SpeedLong before iPods that hold thousands of songs and play them with superb audio quality, individual songs were delivered on 75-rpm and 45-rpm circular records. A 45-rpm record has an angular speed of 45 revolutions per minute. Find the linear speed, in inches per minute, at the point where the needle is 1.5 inches from the record’s center.Before applying the formula we must express in

terms of radians per minute: r

45 revolutions 2 radians1 minute 1 revolution

90 radians

1 minute


Example: Finding Linear Speed (continued)A 45-rpm record has an angular speed of 45 revolutions per minute. Find the linear speed, in inches per minute, at the point where the needle is 1.5 inches from the record’s center.The angular speed of the record is radians per

minute. The linear speed is90

r 90 135 in1.5 inches1 minute min

424 in min


Chapter 5TrigonometricFunctions


5.2 Right Triangle Trigonometry


Objectives:

Use right triangles to evaluate trigonometric functions.

Find function values for

Recognize and use fundamental identities.

Use equal cofunctions of complements.

Evaluate trigonometric functions with a calculator.

Use right triangle trigonometry to solve applied problems.

30 ,45 , and 60 .6 4 3


The Six Trigonometric Functions

The six trigonometric functions are:Function Abbreviationsine sincosine costangent tancosecant cscsecant seccotangent cot


Right Triangle Definitions of Trigonometric Functions

In general, the trigonometric functionsof depend only on the size of angleand not on the size of the triangle.


Right Triangle Definitions of Trigonometric Functions (continued)

In general, the trigonometric functionsof depend only on the size of angleand not on the size of the triangle.


Example: Evaluating Trigonometric FunctionsFind the value of the six trigonometric functions in the figure.We begin by finding c.

2 2 2a b c 2 2 23 4 9 16 25c

25 5c 3sin5

4cos5

3tan4

5csc3

5sec4

4cot3


Function Values for Some Special Angles

A right triangle with a 45°, or radian, angle is isosceles – that is, it has two sides of equal length.

4


Function Values for Some Special Angles (continued)

A right triangle that has a 30°, or radian, angle also has a 60°, or radian angle. In a 30-60-90 triangle, the measure of the side opposite the 30° angle is one-half the measure of the hypotenuse.

6

3


Example: Evaluating Trigonometric Functions of 45°

Use the figure to find csc 45°, sec 45°, and cot 45°.length of hypotenusecsc45

length of side opposite 45

2 2

1

length of hypotenusesec45length of side adjacent to 45

2 21

length of side adjacent to 45cot 45length of side opposite 45

1 11


Example: Evaluating Trigonometric Functions of 30° and 60°

Use the figure to find tan 60° and tan 30°. If a radical appears in a denominator, rationalize the denominator.

length of side opposite 60tan 60length of side adjacent to 60

3 31

length of side opposite 30tan30length of side adjacent to 30

1 1 3 333 3 3


Trigonometric Functions of Special Angles


Fundamental Identities


Example: Using Quotient and Reciprocal Identities

Given and find the value of each of the four remaining trigonometric functions.

2sin3

5cos

3

sintancos

235

3

2 3 23 5 5

2 5 2 555 5

1cscsin

1 32 23


Example: Using Quotient and Reciprocal Identities (continued)

Given and find the value of each of the four remaining trigonometric functions.

2sin3

5cos

3

1seccos

1 35 5

3

3 5 3 555 5

1cottan

1 52 5 2 5

5

5 5 5 5 52 5 22 5 5


The Pythagorean Identities


Example: Using a Pythagorean Identity

Given that and is an acute angle, find the value of using a trigonometric identity.

1sin2

cos

2 2sin cos 1 2

21 cos 12

21 cos 14

2 1cos 14

2 3cos4

3 3cos4 2


Trigonometric Functions and Complements

Two positive angles are complements if their sum is 90° or Any pair of trigonometric functions f and g for which and are called cofunctions.

.2

( ) (90 )f g ( ) (90 )g f


Cofunction Identities


Using Cofunction IdentitiesFind a cofunction with the same value as the given expression:a.

b.

sin 46 cos(90 46 ) cos44

cot12 6 5tan tan tan

2 12 12 12 12


Using a Calculator to Evaluate Trigonometric Functions

To evaluate trigonometric functions, we will use the keys on a calculator that are marked SIN, COS, and TAN. Be sure to set the mode to degrees or radians, depending on the function that you are evaluating. You may consult the manual for your calculator for specific directions for evaluating trigonometric functions.


Example: Evaluating Trigonometric Functions with a Calculator

Use a calculator to find the value to four decimal places:a. sin 72.8° (hint: Be sure to set the calculator to

degree mode)

b. csc 1.5 (hint: Be sure to set the calculator to radian mode)

sin 72.8 0.9553

csc1.5 1.0025


Applications: Angle of Elevation and Angle of Depression

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 the horizontal line and the line of sight to an object that is below the horizontal line is called the angle of depression.


Example: Problem Solving Using an Angle of ElevationThe irregular blue shape in the figure 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?tan 24

750a 750tan 24a

333.9a

The distance across the lakeis approximately 333.9 yards.

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Chapter 5 Trigonometric Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 5.1 Angles and Radian Measure