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Positive Angles
Prepared by Title V Staff:Daniel Judge, Instructor
Ken Saita, Program Specialist
East Los Angeles College
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Generating a positive right angle . . .
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Rotate the initial side counter-clockwise (¼ revolution).
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Generating a positive straight angle . . .
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Rotate the initial side counter-clockwise (½ revolution).
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m() = 180
Why?
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1)Rotate ¼ revolution ccw
2)Rotate another ¼ revolution ccw
You have rotated ½ revolution ccw!
90 + 90 = 180
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Note: Any angle that measures 180 is called a straight angle.
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Rotate the initial side counter-clockwise ¾ revolution.
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So that, m() = 90 + 90 + 90 m() = 270
INITIAL SIDE
TERMINAL SIDE
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Rotate the initial side counter-clockwise 1 revolution
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So that, m() = 90 + 90 + 90 + 90 m() = 360
Note: Initial side = terminal side.
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Q: What would a 45 angle look like?
Answer --
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Q: What would a 30 angle look like?
Answer --
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Note
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Q: What would a 120 angle look like?
Answer --
INITIAL SIDEINITIAL SIDE
TERMINAL SIDE
TERMINAL SIDE
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Note: this procedure can be used to generate the angles 120, 150, 180
210, 240, 270 300, 330, 360.
This is why the system of degrees is based on a circle!
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Q: Can we ever rotate the initial side counterclockwise more than one revolution?
Answer – YES!
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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, . . .
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Picture
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In fact,
rotating the initial side counter-clockwise n revolutions (from 0) generates the angles n 360
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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?
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Answer --
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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.
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1 Revolution --
m() = 30+360m() = 390
390° 1 REV
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2 Revolutions
m() = 30+360+360m() = 30+2360m() = 30+720m() = 750
750° 2 REVS
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3 Revolutions
m() = 30+360+360+360m() = 30+3360m() = 30+1080m() = 1110
1110° 3 REVS
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Q: What if we start at 30 and rotate counterclockwise n revolutions? What angle does this generate?
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Answer --
m() = 30+360n
30°
NOW,
n REV
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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 = + n•360°
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Definition: Coterminal Angles
Angles and are said to be coterminal
if
n360
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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
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End of Positive Angles
Title V East Los Angeles College
1301 Avenida Cesar ChavezMonterey Park, CA 91754
Phone: (323) 265-8784
Email Us At:menteprog@hotmail.com
Our Website:http://www.matematicamente.org
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