Angles. I. Definitions Ray : A part of a line with a single endpoint.
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Transcript of Angles. I. Definitions Ray : A part of a line with a single endpoint.
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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 ∙𝜋180o
radians6 =
30
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120o ∙𝜋180o
120o radians3
= 2
120120o
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225o ∙𝜋180o
225o radians4
= 5
225
225o
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330o ∙𝜋180o
330o radians6
= 11
330330o
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∙180o
𝜋
60𝜋3 ¿60o
60𝝅𝟑
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∙180o
𝜋
305𝜋6 ¿150o
150𝟓𝝅𝟔
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∙180o
𝜋
604𝜋3 ¿240o240
𝟒𝝅𝟑
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∙180o
𝜋
457𝜋4
𝟕𝝅𝟒 315
¿315o
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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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80∘
Find the reference angle.
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65∘
−295∘
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80∘
260∘
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30∘
−69 0∘
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60∘2𝜋3
∙180o
𝜋
602𝜋3 ¿120o
2
11
120o
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IV. Arc Lengths
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𝑠=𝑟 ∙𝜃𝑖𝑛𝑟𝑎𝑑𝑖𝑎𝑛𝑠
q
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𝑠=12 𝑓𝑡 ∙4𝜋3
12 𝑓𝑡
4
1𝑠=16𝜋 𝑓𝑡
𝑠=𝑟 ∙𝜃
4𝜋3𝑟𝑎𝑑
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𝑠=4𝑚∙ 45°
45°
4𝑚
∙𝜋𝑟𝑎𝑑180∘
4
1
𝑠=4𝑚∙𝜋4
𝑠=𝜋𝑚𝑒𝑡𝑒𝑟𝑠
𝑠=𝑟 ∙𝜃