Angles and Their Measure

Section 4.1

Objectives

• I can label the unit circle for radian angles

• I can draw and angle showing correct rotation in Standard Format

• I can calculate Reference Angles

• I can determine what quadrant an angle is in

AnglesAn angle is formed by two rays that have a common endpoint called the vertex. One ray is called the initial side and the other the terminal side. The arrow near the vertex shows the direction and the amount of rotation from the initial side to the terminal side.

Aè

B

C

Terminal Side Initial Side

Vertex

Section 4.1: Figure 4.2, Angle in Standard Position

Section 4.1: Figure 4.3, Positive and Negative Angles

Angles of the Rectangular Coordinate System

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.

x

y

Terminal Side

Initial SideVertex

is positive

Positive angles rotate counterclockwise.

x

y

Terminal Side

Initial SideVertex

is negative

Negative angles rotate clockwise.

Measuring Angles Using DegreesThe figures below show angles classified by

their degree measurement. An acute angle measures less than 90º. A right angle, one quarter of a complete rotation, measures 90º and can be identified by a small square at the vertex. An obtuse angle measures more than 90º but less than 180º. A straight angle has measure 180º.

Acute angle0º < < 90º

90º

Right angle1/4 rotation

Obtuse angle90º < < 180º

180º

Straight angle1/2 rotation

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 MeasureConsider an arc of length s on a circle or radius r.The measure of the central angle that interceptsthe arc is = s/r radians.

O

r

s

r

Section 4.1: Figure 4.6, Illustration of Six Radian

Lengths

Section 4.1: Figure 4.7, Common Radian Angles

Definition of a Reference Angle

• Let be a non-acute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle ´ prime formed by the terminal side or and the x-axis.

Example

a

b

a

b

P(a, b)

Find the reference angle , for the following angle: =315º

Solution:

=360º - 315º = 45º

Example

Find the reference angles for:345

135

6

5

4

11

15345360

45180225360135

66

5

6

6

6

5

44

3

4

32

4

11

Homework

• WS 8-1

• Start learning Unit circle