Pre-Calc Exam Notes 34

Pre-Calc Exam Notes 34 - 34 Chapter 1 Right Triangle...

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34 Chapter 1 Right Triangle Trigonometry §1.5 Rotating an angle θ by 90 in the clockwise direction results in the angle θ 90 . We could use another geometric argument to derive trigonometric relations involving θ 90 , but it is easier to use a simple trick: since formulas (1.4) (1.6) hold for any angle θ , just replace θ by θ 90 in each formula. Since ( θ 90 ) + 90 = θ , this gives us: sin ( θ 90 ) = − cos θ (1.7) cos ( θ 90 ) = sin θ (1.8) tan ( θ 90 ) = − cot θ (1.9) We now consider rotating an angle θ by 180 . Notice from Figure 1.5.4 that the angles θ ± 180 have the same terminal side, and are in the quadrant opposite θ . x y θ + 180 θ 180 ( x , y ) ( x , y ) θ 180 180 r r (a) QI and QIII x y θ + 180 θ 180 ( x , y ) ( x , y ) θ 180 180 r r (b) QII and QIV Figure 1.5.4 Rotation of θ by ± 180 Since ( x , y ) is on the terminal side of θ ± 180 when ( x , y ) is on the terminal side of θ , we get the following relations, which hold for all
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