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Positive Angles Prepared by Title V Staff: Daniel Judge, Instructor Ken Saita, Program Specialist East Los Angeles College EXIT BACKNEXT © 2002 East Los Angeles College. All rights reserved. Click one of the buttons below or press the enter key

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Generating a positive right angle... EXIT BACKNEXT

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Rotate the initial side counter- clockwise (¼ revolution). EXIT BACKNEXT

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Generating a positive straight angle... EXIT BACKNEXT

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Rotate the initial side counter- clockwise (½ revolution). EXIT BACKNEXT

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m( ) = 180 Why? EXIT BACKNEXT

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1)Rotate ¼ revolution ccw 2)Rotate another ¼ revolution ccw You have rotated ½ revolution ccw! 90 + 90 = 180 EXIT BACKNEXT

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Note: Any angle that measures 180 is called a straight angle. EXIT BACKNEXT

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Rotate the initial side counter- clockwise ¾ revolution. EXIT BACKNEXT

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So that, m( ) = 90 + 90 + 90 m( ) = 270 INITIAL SIDE TERMINAL SIDE EXIT BACKNEXT

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Rotate the initial side counter- clockwise 1 revolution EXIT BACKNEXT

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So that, m( ) = 90 + 90 + 90 + 90 m( ) = 360 Note: Initial side = terminal side. EXIT BACKNEXT

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Q: What would a 45 angle look like? Answer -- EXIT BACKNEXT

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Q: What would a 30 angle look like? Answer -- EXIT BACKNEXT

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Note EXIT BACKNEXT

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Q: What would a 120 angle look like? Answer -- INITIAL SIDE TERMINAL SIDE EXIT BACKNEXT

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Note: this procedure can be used to generate the angles120 , 150 , 180 210 , 240 , 270 300 , 330 , 360 . This is why the system of degrees is based on a circle! EXIT BACKNEXT

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Q: Can we ever rotate the initial side counterclockwise more than one revolution? Answer – YES! EXIT BACKNEXT

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Note: Complete Revolutions Rotating the initial side counter- clockwise 1 rev., 2 revs., 3revs.,... generates the angles which measure 360 , 720 , 1080 ,... EXIT BACKNEXT

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Picture EXIT BACKNEXT

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In fact, rotating the initial side counter- clockwise n revolutions (from 0 ) generates the angles n 360 EXIT BACKNEXT

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Q: What if we start at 30 , and now rotate our terminal side 1 complete revolution. What angle did we generate? EXIT BACKNEXT

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Answer -- EXIT BACKNEXT

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What if we start at 30 and now rotate our terminal side counter- clockwise 1 rev., 2 revs., or 3 revs. EXIT BACKNEXT

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1 Revolution -- m( ) = 30 +360 m( ) = 390 390° 1 REV EXIT BACKNEXT

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2 Revolutions m( ) = 30 +360 +360 m( ) = 30 +2 360 m( ) = 30 +720 m( ) = 750 750° 2 REVS EXIT BACKNEXT

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3 Revolutions m( ) = 30 +360 +360 +360 m( ) = 30 +3 360 m( ) = 30 +1080 m( ) = 1110 1110° 3 REVS EXIT BACKNEXT

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Q: What if we start at 30 and rotate counterclockwise n revolutions? What angle does this generate? EXIT BACKNEXT

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Answer -- m( ) = 30 +360 n 30° NOW, n REV EXIT BACKNEXT

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We can generalize this procedure. Let’s start at an angle , then rotate n rev counterclockwise. What formula is generated? NOW, n REV = + n360° EXIT BACKNEXT

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Definition: Coterminal Angles Angles and are said to be coterminal if n 360 EXIT BACKNEXT

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Example– The following angles are coterminal: 0 , 360 , 720 , 1080 ,...coterminal 30 , 390 , 750 , 1110 ,...coterminal 45 , 405 , 765 , 1125 ,...coterminal 60 , 420 , 780 , 1140 ,...coterminal EXIT BACKNEXT

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End of Positive Angles Title V East Los Angeles College 1301 Avenida Cesar Chavez Monterey Park, CA 91754 Phone: (323) 265-8784 Email Us At: menteprog@hotmail.com Our Website: http://www.matematicamente.org EXIT BACKNEXT

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13.2 Angles of Rotation and Radian Measure

13.2 Angles of Rotation and Radian Measure

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