13 2 Radian and Degree Measure In this

13. 2 Radian and Degree Measure In this section, we will study the following topics: n n n Terminology used to describe angles Degree measure of an angle Radian measure of an angle Converting between radian and degree measure Find coterminal angles 1

13. 2 Radian and Degree Measure Angles Trigonometry: measurement of triangles Angle Measure 2

13. 2 Radian and Degree Measure Standard Position Vertex at origin The initial side of an angle in standard position is always located on the positive x-axis. 3

13. 2 Radian and Degree Measure Positive and negative angles When sketching angles, always use an arrow to show direction. 4

13. 2 Radian and Degree Measuring Angles The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. There are two common ways to measure angles, in degrees and in radians. We’ll start with degrees, denoted by the symbol º. One degree (1º) is equivalent to a rotation of of one revolution. 5

13. 2 Radian and Degree Measuring Angles 6

13. 2 Radian and Degree Measure Classifying Angles are often classified according to the quadrant in which their terminal sides lie. Ex 1: Name the quadrant in which each angle lies. 50º Quadrant 1 208º Quadrant 3 II I -75º Quadrant 4 III IV 7

13. 2 Radian and Degree Measure Classifying Angles Standard position angles that have their terminal side on one of the axes are called quadrantal angles. For example, 0º, 90º, 180º, 270º, 360º, … are quadrantal angles. 8

13. 2 Radian and Degree Measure Coterminal Angles that have the same initial and terminal sides are coterminal. Angles and are coterminal. 9

13. 2 Radian and Degree Measure Example of Finding Coterminal Angles You can find an angle that is coterminal to a given angle by adding or subtracting multiples of 360º. Ex 2: Find one positive and one negative angle that are coterminal to 112º. For a positive coterminal angle, add 360º : 112º + 360º = 472º For a negative coterminal angle, subtract 360º: 112º - 360º = -248º 10

Ex 3. Find one positive and one negative angle that is coterminal with the angle = 30° in standard position. Ex 4. Find one positive and one negative angle that is coterminal with the angle = 272 in standard position.

13. 2 Radian and Degree Measure Radian Measure A second way to measure angles is in radians. Definition of Radian: One radian is the measure of a central angle that intercepts arc s equal in length to the radius r of the circle. In general, 12

13. 2 Radian and Degree Measure Radian Measure 13

13. 2 Radian and Degree Measure Radian Measure 14

13. 2 Radian and Degree Measure Conversions Between Degrees and Radians 1. To convert degrees to radians, multiply degrees by 2. To convert radians to degrees, multiply radians by 15

Ex 5. Convert the degrees to radian measure. a) 60 b) 30 c) -54 d) -118 e) 45

Ex 6. Convert the radians to degrees. a) b) c) d)

Ex 7. Find one positive and one negative angle that is coterminal with the angle = in standard position. Ex 8. Find one positive and one negative angle that is coterminal with the angle = in standard position.

Degree and Radian Form of “Special” Angles 90 ° 120 ° 60 ° 135 ° 45 ° 150 ° 30 ° 0° 180 ° 360 ° 210 ° 330 ° 225 ° 315 ° 240 ° 300 ° 270 ° 19

Common Degrees/Radians 0º 0 135º 270º 30º 150º 300º 45º 180º 315º 60º 210º 330º 90º 225º 360º 120º 240º

Class Work Convert from degrees to radians. 1. 54 2. -300 Convert from radians to degrees. 3. 4.

Find one postive angle and one negative angle in standard position that are coterminal with the given angle. 5. 135 6.