Trigonometric Functions Angles and Radian Measures

Trigonometric FunctionsAngles and Radian Measures

Objectives

Identifying the parts of an angle Measuring angles using degrees and radians Calculating radian measure Conversion between degrees and radians Drawing angles in standard position Finding coterminal angles Calculating the length of a circular arc

Vocabulary

Acute angleAngleCoterminal anglesInitial sideNegative angleObtuse anglePositive angleQuadrantal angleRadianRadian Measure

RayReflex angleRight angleQuadrantTerminal sideVertex

AnglesA ray is a part of a line that has only one endpoint and extends forever in one direction.

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

Initial side

Term

inal s

ide

An arrow near the vertex shows the direction and the amount of rotation from the initial side to the terminal side. Several methods can be used to name an angle.

A

B

C1

By using the end points with the vertex in the middle

L CAB L BAC

By using the letter of the vertex or a number

L A L 1

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.

There are two types of rotation: (1) a counterclockwise rotation which results in a positive angle and (2) a clockwise rotation which yields a negative angle.

When an angle is in standard position, its terminal side can lie in a quadrant. If the terminal side lies on the x-axis or y-axis, it does not lie in any quadrant. In this case it is called a quadrantal angle.

Positive x-axisInitial side

Terminal side

Measuring Angles Using Degrees

We use the symbol º for degrees to indicate the measure of an angle. Certain angles have certain names based on their measures.

0 °<𝜃<90 °

𝜃=90 °

90 °<𝜃<180 °

𝜃=180 °

180 °<𝜃<360 °

Radian Measure

Consider an arc of length s on a circle of radius r. The measure of the central angle, θ, that intercepts the arc is

A central angle in a circle with a radius of 6 inches intercepts an arc of length 15 inches. What is the radian measure of the central angle?

Relationship Between Degrees and Radians

We know that 360 degrees is the amount of rotation of a ray back onto itself. Therefore we can come up with the following:

One complete rotation measures 360º.

One complete rotation measures radians

360º = radians 180º = radians

Conversion Between Degrees and Radians

Using the basic relationship radians = 180º,

1. To convert degrees to radians, multiply degrees by

2. To convert radians to degrees, multiply radians by

Angles that are fractions of a complete rotation are usually expressed in radian measure as fractional multiples of instead of decimal approximations.

We write rather than

Convert each angle in degrees to radians:

𝜋6

𝜋2

−3𝜋4

30 °∗( 𝜋180 ° )=¿

Remember:

90 °∗( 𝜋180 ° )=¿

−135 °∗( 𝜋180

°)=¿

Convert each angle in radians to degrees:

radians

radians

radian

60 °

300 °

57.32 °

𝜋=180 °=3.14 radians𝜋3∗

180 °𝜋

=¿

5𝜋3∗

180 °𝜋

=¿

1 radian∗180 °

3.14 radians

Degree and Angle Measures of Selected Positive and Negative Angles

Positive angles Negative angles

Drawing Angles in Standard Position

Draw and label each angle in standard position:

Which quadrant does each angle lie in?

Quadrant I

Quadrant III

Quadrant III

Quadrant IV

III

III IV

Coterminal AnglesTwo angles with the same initial and terminal sides but possibly different rotations are called coterminal angles. To find an angle that is coterminal, use the following formulas:

, where k is an integer or , where k is an integer

K is the number of rotations.

Initial side

Term

inal

sid

e

Initial side

Term

inal

sid

e

Find a positive angle that is less than 360º that is coterminal with each of the following:

420º angle

− 120º angle

400º angle

− 135º angle

60 °

240 °

40 °

225 °

, where k is an integer

420 °−360 °=¿

−120 °+360 °=¿

400 °−360 °=¿

−135+360=¿

Find a positive angle that is less than that is coterminal with each of the following:

angle

− angle

angle

− angle

5𝜋6

23𝜋12

3𝜋5

29𝜋15

, where k is an integer

17𝜋6−2𝜋=¿

17𝜋6−

12𝜋6

=¿

−𝜋12

+2𝜋=¿−𝜋12

+24 𝜋

12=¿

13𝜋5−2𝜋=¿

13𝜋5−

10𝜋5

=¿

−𝜋15

+2𝜋=¿−𝜋15

+30𝜋15

=¿

The Length of a Circular Arc

The length of the arc intercepted by the central angle is

A circle has a radius of 10 inches. Find the length of the arc intercepted by a central angle of 120º.

Note: the angle, , must be converted to radians. Then plug into the equation.

𝜃=120 °∗𝜋

180°=

2𝜋3

𝑠=10∗2𝜋3

=20𝜋

3=

20∗3.143

≈20.93 inches

Linear speed is defined as the distance traveled for a given time.

s is the arc length r is the radiust is the time is the angle expressed in radians

𝒗=𝒔𝒕

=𝒓 𝜽𝒕

Linear Speed

▪ The second hand of a clock is 8 centimeters long. Find the linear speed of the tip of this second hand.

360 2

8(2 ) 1 minute

1minute 60 seconds

.8378cm/sec

s rv

t t

v

v

Angular speed is defined as the angle covered for a given time.

the angle measured in radianst time

Angular Speed

The circular blade on a saw rotates at 2400 revolutions per minute. Find the angular speed in radians per second.

1 rev 2 radians so

2400 rev = 4800 radians

4800 1 minute

1 minute 60 seconds80 radians/sec

wt

w

v