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Angles
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I. Definitions
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Ray : A part of a line with a single endpoint
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Angle : Two rays with a common endpoint
Initial Side
Terminal Side
Vertex
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QUADRANTS IIIIVIII
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An angle with its vertex at the origin and its initial side on the positive x-axis is in standard form.
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Positive Angles: Angles measured from the initial side counterclockwise to its terminal side.
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Negative Angles: Angles measured from the initial side clockwise to its terminal side.
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II. Measuring Angles
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Acute Angle : Angles between 0 and 90o
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Right Angle : Angles measuring 90o
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Obtuse Angle : Angles between 90o and 180o
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Straight Angle : Angles that measure 180o
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The size of an angle is determined by the space between the two sides (rays).These are measured in two different units:
Degrees Radians
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Degrees Radians360o 2p
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Degrees Radians180o p
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So we can concluded180o = p radians
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III. Converting between Radians and Degrees
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30o30o βπ180o
radians6 =
30
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120o βπ180o
120o radians3
= 2
120120o
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225o βπ180o
225o radians4
= 5
225
225o
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330o βπ180o
330o radians6
= 11
330330o
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β180o
π
60π3 ΒΏ60o
60π π
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β180o
π
305π6 ΒΏ150o
150ππ π
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β180o
π
604π3 ΒΏ240o240
ππ π
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β180o
π
457π4
ππ π 315
ΒΏ315o
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IV. Reference & Co-terminal Angles
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Reference angle: the acute angle measured from the terminal side to the nearest x-axis.
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80β
Find the reference angle.
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65β
β295β
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80β
260β
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30β
β69 0β
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60β2π3
β180o
π
602π3 ΒΏ120o
2
11
120o
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IV. Arc Lengths
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π =π βππππππππππ
q
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π =12 ππ‘ β4π3
12 ππ‘
4
1π =16π ππ‘
π =π βπ
4π3πππ
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π =4πβ 45Β°
45Β°
4π
βππππ180β
4
1
π =4πβπ4
π =ππππ‘πππ
π =π βπ