TRIGONOMETRY

Trigonometry Comes from two Greek words trigonon and metron, meaning “triangle measurement”

7.1 Measurement of Angles

DEGREES-unit of measure for angles360°=1 revolution

Revolution –one complete circular motion

Minutes- 1°=60 minutes (60’)

Seconds-1’ =60 seconds(60”)1°=3600”

Convert to degrees only

25 20'6"

20 625

60 3600 25.335

Convert to degrees only

40 53'12"

53 1240

60 3600 40.887

Convert to DMS 12.36.36 60 21.6

12 21'36"

.6 60 36

Convert to DMS 85.78.78 60 46.8

85 46 '48"

.8 60 48

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

O

r

s

r

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-unit of measure for angles2π=1 revolution

Converting degrees to radians

πdegrees× =radians

180

Converting radians to degrees

180radians× =degrees

π

Example 240°

240240

180 180

4

3

Example -315°7

4

Example

11 180 1980

6 6

330

135 Example

11

6

3

4

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.

Aθ

B

Terminal Side Initial Side

Vertex

C

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.

Coterminal Angles

An angle of xº is coterminal (have the same terminal ray) with angles of

xº + k · 360ºwhere k is an integer.

Assume the following angles are in standard position. Find a positive angle less than 360º that is coterminal with: a. a 420º angle b. a –120º angle.

Solution We obtain the coterminal angle by adding or subtracting 360º.

Our need to obtain a positive angle less than 360º determines whether we should add or subtract. a. For a 420º angle, subtract 360º to find a positive coterminal angle.

420º – 360º = 60ºA 60º angle is coterminal with a 420º angle. These angles, shown on

the next slide, have the same initial and terminal sides.

Example

Solution b. For a –120º angle, add 360º to find a positive coterminal angle.

-120º + 360º = 240ºA 240º angle is coterminal with a –120º angle. These angles have the same initial and terminal sides.

x

y

420º

60º

x

y

240º

-120º

Example cont.

Examples- find two angles, one positive and one negative, that are coterminal with each given angle

1. 10°

2. 100°

3. -5°

4. 400°

5. π

6. π/2

7. -π/3

8. 4π

1. 370°,-350°

2. 460°,-260°

3. 355°,-365°

4. 40°,-320°

5. 3π,-π

6. 5π/2,-3π/2

7. 5π/3,-7π/3

8. ±2π

ASSIGNMENT

•Page 261 #1-10, 12-32 (even)