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**5.2 Understanding Angles terminal arm q initial arm standard position**

non-standard position x y x y terminal arm q q q initial arm 0º

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y Positive angles ex: 80º rotate counterclockwise q x x y Negative angles rotate clockwise q ex: –120º

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**Quadrants II I III IV x y quadrant angle I II III IV**

90º II I quadrant angle I II III IV 0º < q < 90º 180º 90º < q < 180º 0º 180º < q < 270º III IV 270º < q < 360º 270º

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x y Example: P(x, y) Let P(x, y) be a point on the terminal arm of an angle in standard position. q1 Point P can be anywhere in the x-y plane. q1 is in quadrant II. 90º < q1 < 180º x y 270º < q2 < 360º q2 q2 is in quadrant IV. P(x, y)

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x y q3 = 180º P(x, y) q3 q3 lies in the negative x-axis.

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Coterminal angles x y They share the same initial arm and the same terminal arm. q1 Ex: q1 = 210º q2 q2 = – 150º

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x y Coterminal angles share the same initial arm and the same terminal arm. q2 Ex: q1 = 30º q1 q2 = 360º + 30º q3 = 390º q3 = 720º + 30º x y = 750º q4 = – 360º + 30º = – 330º q4

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**The principal angle is the angle between 0º and 360º.**

The coterminal angles of 390º, 750º and – 330º all share the same principal angle of 30º. x y The related acute angle is the angle formed by the terminal arm of an angle in standard position and the x-axis. q1 b The related acute angle lies between 0º and 90º.

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x y x y q q b b q = 220º b = 220º – 180º q = 150º = 40º b = 180º – 150º x y = 30º q = 325º b = 360º – 325º b q = 35º

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**b) determine the principal angle. **

Example 1: x y a) Sketch an angle of –160º b) determine the principal angle. principal angle c) determine the related acute angle. related acute angle a) –160º will be in quadrant III. – 160º b) The principal angle: 360º– 160º = 200º c) The related acute angle: 200º– 180º = 20º

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**b) Determine the value of the related acute angle. q**

Example 2: The point P(–5, –4) lies on the terminal arm an angle in standard position. x y a) Sketch the angle. b) Determine the value of the related acute angle. q 5 b 4 P(–5, –4) c) Determine the principal angle q. q = 180º + 39º q = 219º

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**b) Determine the value of the related acute angle. q**

Example 3: The point P(– 6, 7) lies on the terminal arm an angle in standard position. x y P(–6, 7) a) Sketch the angle. b) Determine the value of the related acute angle. 7 q b 6 c) Determine the principal angle q. q = 180º – 49.4º q = 130.6º

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**Example 4: State all values of q , where n Î I.**

q = 25º + 360ºn, 1 £ n £ 3 q1 = 25º + 360º(1) q2 = 25º + 360º(2) q3 = 25º + 360º(3) = 385º q2 = 25º + 720º q3 = 25º º = 745º = 1105º Answer: {385º, 745º, 1105º }

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