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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