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

Generating a positive right angle... EXIT BACKNEXT

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

Generating a positive straight angle... EXIT BACKNEXT

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

m(  ) = 180  Why? EXIT BACKNEXT

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

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

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

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

Rotate the initial side counter- clockwise 1 revolution EXIT BACKNEXT

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

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

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

Note EXIT BACKNEXT

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

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

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

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

Picture EXIT BACKNEXT

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

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

Answer -- EXIT BACKNEXT

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

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

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

3 Revolutions m(  ) = 30  +360  +360  +360  m(  ) = 30  +3  360  m(  ) = 30   m(  ) = 1110   1110° 3 REVS EXIT BACKNEXT

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

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

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

Definition: Coterminal Angles Angles  and  are said to be coterminal if     n  360  EXIT BACKNEXT

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

End of Positive Angles Title V East Los Angeles College 1301 Avenida Cesar Chavez Monterey Park, CA Phone: (323) Us At: Our Website: EXIT BACKNEXT