Bellwork

Radian and Degree Measure Section 9.2

In this section, we will study the following topics: MAT 200 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

*(Take Note)

Angles Trigonometry: measurement of triangles Angle Measure

(Take note) Standard Position: An angle is in standard position when its vertex is at the origin and its initial side lies on the positive x-axis. Vertex at origin The initial side of an angle in standard position is always located on the positive x-axis.

*(Take Note)Radian and Degree Measure Positive and negative angles When sketching angles, always use an arrow to show direction.

Drawing Angles in Standard Position

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.

Angles  and  are coterminal. (Take Notes) Coterminal Angles Angles that have the same initial and terminal sides are coterminal. Angles  and  are coterminal.

Find one positive angle and one negative angle that are coterminal with (a) −45° and (b) 395°. There are many such angles, depending on what multiple of 360° is added or subtracted.

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 Measure A second way to measure angles is in radians. (Take Note) Radian Measure A second way to measure angles is in radians. Definition of Radian: One radian is the measure of an angle in standard position whose terminal side intercepts an arc of length r. In general,

Radian and Degree Measure Radian Measure

(Take Note) Radian Measure

Take Note

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)

Bellwork Convert the degrees to radian measure. 1. 30 -54 MAT 200 Bellwork Convert the degrees to radian measure. 1. 30 -54 Convert the radians to degrees. 3. 4.

Take Note

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

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

Bellwork Convert from degrees to radians. 54 -300 Convert from radians to degrees. 3. 4.

A sector is a region of a circle that is bounded by two radii and an arc of the circle. The central angle θ of a sector is the angle formed by the two radii. There are simple formulas for the arc length and area of a sector when the central angle is measured in radians.

A softball field forms a sector with the dimensions shown A softball field forms a sector with the dimensions shown. Find the length of the outfield fence and the area of the field.

In Exercises 33–38, use a calculator to evaluate the trigonometric function.