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

Terminal Side  Vertex Initial Side 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.

 Positive Angle  Negative angle

if the rotation is in a counterclockwise direction if the rotation is in a clockwise direction Standard Position

Terminal Side

  45

Vertex Initial Side   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.

Quadrant II

0 180 or -180 Quadrantal Angle is an angle with a terminal side coincided with one of the axis.

Quadrant III

270 or -90 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°.

Example 2 Give the angle measure represented by each rotation. b. 4.2 rotations counterclockwise a.3.75 rotations clockwise 4.2 ( 360) = 1512 3.75 (-360) = -1350 Counterclockwise Clockwise rotations rotations have positive have negative measure. measure.

The angle measure of 3.75 The angle measure of 4.2 clockwise rotations is counterclockwise rotations is -1350°. 1512°.

If α is the degree measure of the angle then α +360 k, where k is an integer, are coterminal with α. a. 42° b. 128° All angles having a measure All angles having a measure of 42° + 360k°, where k is an of 128° + 360k°, where k is integer, are coterminal with an integer, are coterminal 42°. with 128°.

A positive angle is A positive angle is 42° + 360°(1) or 402°. 28° + 360°(3) or 1208°. A negative angle is A negative angle is 128° + 360°(-1) or -232°. 42° + 360°(-2) or -678°.  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. α = 85° b. -2408° The angle is -2408°, but the coterminal angle needs to be positive.

The coterminal angle (α) is 112°. Its terminal side lies in the second quadrant. 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 a. 240° b. -305° Since 240° is between A coterminal angle of 180° and 270°, the -305° is 360° - 305° or 55°. Since 55° is between terminal side of the 0° and 90°, the terminal angle is in the third side of the angle is in quadrant. the first quadrant.

240° - 180° = 60°

The reference angle is The reference angle is 60° 55°