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TRIGONOMETRY. Trigonometry Comes from two Greek words trigonon and metron, meaning “triangle measurement”. 7.1 Measurement of Angles. DEGREES-unit of measure for angles 360 °=1 revolution Revolution –one complete circular motion Minutes- 1°=60 minutes (60’)

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trigonometry

TRIGONOMETRY

Trigonometry

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

slide3

DEGREES-unit of measure for angles

360°=1 revolution

Revolution –one complete circular motion

Minutes- 1°=60 minutes (60’)

Seconds-1’ =60 seconds(60”)

1°=3600”

radian measure
Consider an arc of length s on a circle of radius r.

The measure of the central angle that intercepts

the arc is

 = s/r radians.

s

r

O

r

Radian Measure
definition of a radian
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 angles 2π=1 revolution

slide10

Converting degrees to radians

Converting radians to degrees

example 240
Example 240°

Example -315°

example
Example

Example

angles

C

A

Initial Side

Terminal Side

θ

Vertex

B

Angles

An 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 terminalside. The arrow near the vertex shows the direction and the amount of rotation from the initial side to the terminal side.

angles of the rectangular coordinate system

y

y

is positive

Terminal Side

Vertex

Initial Side

x

x

Vertex

Terminal Side

Initial Side

is negative

Positive angles rotate counterclockwise.

Negative angles rotate clockwise.

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

xº+ k· 360º

where k is an integer.

Coterminal Angles
example16

Example

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 = 60º

A 60º angle is coterminal with a 420º angle. These angles, shown on the next slide, have the same initial and terminal sides.

example cont

y

y

240º

60º

x

x

420º

-120º

Example cont.

Solution

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

-120º + 360º = 240º

A 240º angle is coterminal with a –120º angle. These angles have the same initial and terminal sides.

examples find two angles one positive and one negative that are coterminal with each given angle
10°

100°

-5°

400°

π

π/2

-π/3

370°,-350°

460°,-260°

355°,-365°

40°,-320°

3π,-π

5π/2,-3π/2

5π/3,-7π/3

±2π

Examples- find two angles, one positive and one negative, that are coterminal with each given angle
assignment
ASSIGNMENT
  • Page 261 #1-10, 12-32 (even)