Radian and Degree Measure MAT 200 Radian and Degree Measure In this section, we will study the following topics: Terminology used to describe angles Degree measure of an angle Radian measure of an angle Converting between radian and degree measure Find coterminal angles Reference angles

Radian and Degree Measure Angles Trigonometry: measurement of triangles Angle Measure

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.

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

Radian and Degree Measure 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.

Radian and Degree Measure Measuring Angles

Radian and Degree Measure Classifying Angles Angles are often classified according to the quadrant in which their terminal sides lie. Ex1: Name the quadrant in which each angle lies. 50º 208º II I -75º III IV Quadrant 1 Quadrant 3 Quadrant 4

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.

Radian and Degree Measure Coterminal Angles Angles that have the same initial and terminal sides are coterminal. Angles  and  are coterminal.

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

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.

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,

Radian and Degree Measure Radian Measure

Radian and Degree Measure Radian Measure

Radian and Degree Measure Conversions Between Degrees and Radians To convert degrees to radians, multiply degrees by To convert radians to degrees, multiply radians by

Ex 5. Convert the degrees to radian measure. 60 30 -54 -118 45

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

MAT 200 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 0°  360 °  30 °  45 °  60 °  330 °  315 °  300 °   120 °  135 °  150 °  240 °  225 °  210 °  180 ° 90 °  270 °   Degree and Radian Form of “Special” Angles

Reference Angles

Reference Angles The values of the trigonometric functions of angles greater than 90  (or less than 0 ) can be determined from their values at corresponding acute angles called reference angles.

Reference Angles Figure 4.37 shows the reference angles for  in Quadrants II, III, and IV. Figure 4.37

Example 4 – Finding Reference Angles Find the reference angle  . a.  = 300  b.  = 2.3 c.  = –135  Solution: a. Because 300  lies in Quadrant IV, the angle it makes with the x-axis is   = 360  – 300  = 60 . Degrees

Example 4 – Solution cont’d b. Because 2.3 lies between  /2  1.5708 and   3.1416, it follows that it is in Quadrant II and its reference angle is   =  – 2.3  0.8416. c. First, determine that –135  is coterminal with 225 , which lies in Quadrant III. So, the reference angle is   = 225  – 180  = 45 . Radians Degrees

Example – Finding Reference Angles Find the reference angle  ′. a.  = 300 b.  = 2.3 c.  = –135

Example (a) – Solution Because 300 lies in Quadrant IV, the angle it makes with the x-axis is  ′ = 360 – 300 = 60. The figure shows the angle  = 300 and its reference angle  ′ = 60. Degrees

Example (b) – Solution cont’d Because 2.3 lies between  /2  1.5708 and   3.1416, it follows that it is in Quadrant II and its reference angle is  ′ =  – 2.3  0.8416. The figure shows the angle  = 2.3 and its reference angle  ′ =  – 2.3. Radians

Example (c) – Solution cont’d First, determine that –135 is coterminal with 225, which lies in Quadrant III. So, the reference angle is  ′ = 225 – 180 = 45. The figure shows the angle  = –135 and its reference angle  ′ = 45. Degrees

Your Turn: Find the reference angle for each of the following. 213° 1.7 rad −144° -144 ̊ is coterminal to 216 ̊ 216 ̊ - 180 ̊ = 36 ̊

Example 4 – Solution cont’d Figure 4.38 shows each angle  and its reference angle  . (a) (b) (c) Figure 4.38

Reference Angles Review Reference Angle: the smallest positive acute angle determined by the x-axis and the terminal side of θ ref angle ref angle ref angle ref angle Think of the reference angle as a “distance”—how close you are to the closest x-axis.

Reference Angles When your angle is negative or is greater than one revolution, to find the reference angle, first find the positive coterminal angle between 0° and 360° or 0 and 2𝜋.

Class Work Convert from degrees to radians. 54 -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. 135

Video Link www.youtube.com/watch?v=wcfkDuFpbiM www.youtube.com/watch?v=6BRtPfofXog https://www.khanacademy.org/math/algebra2/trig-functions/intro-to-radians-alg2/v/we-converting-degrees-to-radians www.youtube.com/watch?v=qgGFQXnuszk