Angles and Degree Measures

VertexInitial Side

Terminal Side

An angle may be generated by rotating one of two rays that share a fixed endpoint, a vertex.

Positive Angle Negative angle

if the rotation is ina counterclockwise direction if the rotation is in

a clockwise direction

The measure of an angle provides us with information concerning the direction of the rotation and the amount of the rotation necessary to move from the initial side of the angle to the terminal side.

Standard Position

Vertex Initial Side

Terminal Side

45

45

Quadrantal Angle is an angle with a terminal side coincided with one of the axis.

0 or -360

Quadrantal Angle is an angle with a terminal side coincided with one of the axis.

90 or -270

Quadrant I

Quadrantal Angle is an angle with a terminal side coincided with one of the axis.

0180 or -180

Quadrant II

Quadrantal Angle is an angle with a terminal side coincided with one of the axis.

270 or -90

Quadrant III

Quadrantal Angle is an angle with a terminal side coincided with one of the axis.

360 or 0

Quadrant VI

Degree measurement

came from the

Babylonian culture.

They used a numerical system

based on 60 rather than 10.

They assumed that the

measure of each angle of

an equilateral triangle is 60

The 1/60 of the measure of each angle of an equilateral triangle is 1degree

The 1/60 of 1degree is 1’( minute).

The 1/60 of 1’ is 1”(second)

Example 1

GEOGRAPHY Geographic locations are typically expressed

in terms of latitude and longitude.

a. Las Vegas, Nevada, is located at about 36.175°

north latitude. Change to 36.175° to degrees, minutes,

and seconds.

36.175°= 36° + (0.175 60) Multiply the decimal portion of the degree

= 36° + 10.5 measure by 60 to find the number of minutes.

= 36° + 10 + (0.5 60) Multiply the decimal portion of the minute

= 36° + 10 + 30 measure by 60 to find the number of seconds

.

36.175° can be written as 36° 10 30.

b. Las Vegas is also located at 115° 8 11 west longitude.

Write 115 ° 8 11 as a decimal rounded to the nearest

thousandth.

115 ° 8 11= 115° + 8(1/60) + 11(1/3600)

or about 115.136°

115 ° 8 11 can be written as 115.136°.

The angle measure of 3.75 clockwise rotations is -1350°.

Example 2Give the angle measure represented by each rotation.

a.3.75 rotations clockwise

3.75 (-360) = -1350

Clockwise rotations have negative measure.

The angle measure of 4.2 counterclockwise rotations is 1512°.

b. 4.2 rotationscounterclockwise

4.2 ( 360) = 1512

Counterclockwise rotations have positivemeasure.

If α is the degree measure of the angle then α +360 k, where k is an integer, are coterminal with α.

a. 42°All angles having a measure

of 42° + 360k°, where k is an

integer, are coterminal with

42°.

A positive angle is

42° + 360°(1) or 402°.

A negative angle is

42° + 360°(-2) or -678°.

b. 128°All angles having a measureof 128° + 360k°, where k is an integer, are coterminalwith 128°.

A positive angle is28° + 360°(3) or 1208°.

A negative angle is 128° + 360°(-1) or -232°.

α = 85°

a. 445°

In α + 360k°, you need

to find the value of α.

First, determine the

number of complete

rotations (k) by

dividing

445 by 360.

The coterminal angle (α) is 112°. Its terminal side lies in the second quadrant.

b. -2408°

The angle is -2408°, but

the coterminal angle needs to be positive.

For any angle α , 0< α<360, its reference

angle α‘ is defined by

a. α, when the terminal side is in Quadrant I

b. 180- α, when the terminal side is in Quadrant II

c. α-180, when the terminal side is in Quadrant III

d. 360 - α, when the terminal side is in Quadrant IV

The reference angle is 60°

The reference angle is 55°

a. 240°

Since 240° is between

180° and 270°, the

terminal side of the

angle is in the third

quadrant.

240° - 180° = 60°

b. -305°A coterminal angle of -305° is 360° - 305° or55°. Since 55° is between0° and 90°, the terminalside of the angle is inthe first quadrant.