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Angles and Their Measure 4.1

Angles and Their Measure 4.1. C. A. è. Initial Side. Terminal Side. B. Vertex. Angles.

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Angles and Their Measure 4.1

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  1. Angles and Their Measure4.1

  2. C A è Initial Side Terminal Side B Vertex Angles An angle is formed by two rays that have a common endpoint called the vertex. One ray is called the initial side and the other the terminal side. The arrow near the vertex shows the direction and the amount of rotation from the initial side to the terminal side.

  3. y y is positive Terminal Side  Vertex Initial Side x x Vertex  Terminal Side Initial Side is negative Positive angles rotate counterclockwise. Negative angles rotate clockwise. Angles of the Rectangular Coordinate System • An angle is in standard position if • its vertex is at the origin of a rectangular coordinate system and • its initial side lies along the positive x-axis.

  4. 180º  90º  Acute angle 0º <  < 90º Right angle 1/4 rotation Obtuse angle 90º <  < 180º Straight angle 1/2 rotation Measuring Angles Using Degrees The figures below show angles classified by their degree measurement. An acute angle measures less than 90º. A right angle, one quarter of a complete rotation, measures 90º and can be identified by a small square at the vertex. An obtuse angle measures more than 90º but less than 180º. A straight angle has measure 180º.

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

  6. Example Assume the following angles are in standard position. Find a positive angle less than 360º that is coterminal with: a. a 420º angle b. a –120º angle. Solution We obtain the coterminal angle by adding or subtracting 360º. Our need to obtain a positive angle less than 360º determines whether we should add or subtract. a. For a 420º angle, subtract 360º to find a positive coterminal angle. 420º – 360º = 60º A 60º angle is coterminal with a 420º angle. These angles, shown on the next slide, have the same initial and terminal sides.

  7. y y 240º 60º x x 420º -120º Example cont. Solution b. For a –120º angle, add 360º to find a positive coterminal angle. -120º + 360º = 240º A 240º angle is coterminal with a –120º angle. These angles have the same initial and terminal sides.

  8. Example • Find and graph positive and negative coterminal angle for 175 degrees. 535 degrees -185 degrees

  9. Finding Complements and Supplements • For an xº angle, the complement is a 90º – xºangle. Thus, the complement’s measure is found by subtracting the angle’s measure from 90º. • For an xº angle, the supplement is a 180º – xºangle. Thus, the supplement’s measure is found by subtracting the angle’s measure from 180º.

  10. 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.

  11. 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. s  r O r Radian Measure

  12. Conversion between Degrees and Radians • Using the basic relationship  radians = 180º, • To convert degrees to radians, multiply degrees by ( radians) / 180 • To convert radians to degrees, multiply radians by 180 / ( radians)

  13. Convert each angle in degrees to radians 40º 75º -160º Example

  14. Example cont. Solution: • 40º = 40*/180 = 2  /9 • 75º = 75*  /180 = 5  /12 • -160º = -160* /180 = -8  /9

  15. Convert to degrees 180 degrees 45 degrees 216 degrees 105 degrees

  16. 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  s  O r Length of a Circular Arc

  17. 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 Example

  18. A clock has a minute hand that is 3 inches long. Find the distance the tip traveled in 10 minutes.

  19. 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 Definitions of Linear and Angular Speed

  20. Example • A wind machine used to generate electricity has blades that are 10 feet in length. The propeller is rotating at 4 revolutions per second. Find the linear speed, in feet per second, of the tips of the blades. First we find angular speed. The angle is 2pi and it does 4 revolutions in 1 sec. Next we find linear speed by multiplying angular speed times radius

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