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Angles and Their Measure

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Angles and Their Measure

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    1. Angles and Their Measure

    2. Angles

    3. Angles of the Rectangular Coordinate System

    4. Measuring Angles Using Degrees

    5. Coterminal Angles An angle of xş is coterminal with angles of xş + k · 360ş where k is an integer.

    6. Text Example

    7. Text Example cont.

    8. Finding Complements and Supplements For an xş angle, the complement is a 90ş – xş angle. Thus, the complement’s measure is found by subtracting the angle’s measure from 90ş. For an xş angle, the supplement is a 180ş – xş angle. Thus, the supplement’s measure is found by subtracting the angle’s measure from 180ş.

    9. Definition of a Radian One radian is the measure of the central angle of a circle that intercepts an arc equal in length to the radius of the circle.

    10. Radian Measure Consider an arc of length s on a circle or radius r. The measure of the central angle that intercepts the arc is ? = s/r radians.

    11. Conversion between Degrees and Radians Using the basic relationship ? radians = 180ş, To convert degrees to radians, multiply degrees by (? radians) / 180? To convert radians to degrees, multiply radians by 180? / (? radians)

    12. Example Convert each angle in degrees to radians 40ş 75ş -160ş

    13. Example cont. Solution: 40ş = 40*?/180 = 2 ? /9 75ş = 75* ? /180 = 5 ? /12 -160ş = -160* ? /180 = -8 ? /9

    14. Length of a Circular Arc Let r be the radius of a circle and ? the non- negative radian measure of a central angle of the circle. The length of the arc intercepted by the central angle is s = r ?

    15. Example A circle has a radius of 7 inches. Find the length of the arc intercepted by a central angle of 2?/3 Solution: s = (7 inches)*(2 ? /3) =14 ? /3 inches

    16. Definitions of Linear and Angular Speed If a point is in motion on a circle of radius r through an angle of ? radians in time t, then its linear speed is v = s/t where s is the arc length given by s = r ?, and its angular speed is ? = ?/t

    17. Linear Speed in Terms of Angular Speed The linear speed, v, of a point a distance r from the center of rotation is given by v=r? where ? is the angular speed in radians per unit of time.

    18. Angles and Their Measure

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