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Trigonometry Lecture Notes_part2

# 2 example 60 if 5 sin 13 θ = and θ lies in quadrant

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Unformatted text preview: 2. Example 60 If 5 sin 13 θ = and θ lies in quadrant II, find the exact value of: a. sin 2 θ b. cos2 θ c. tan 2 θ Example 61 Find the exact value of 2 2tan15 1 tan 15 °- ° Three Forms of the Double-Angle Formula for cos2 θ 2 2 2 2 cos 2 cos sin cos 2 2cos 1 cos 2 1 2sin θ θ θ θ θ θ θ =- =- = - Example 62 Verify the identity: 3 cos3 4cos 3cos θ θ θ =- Power-Reducing Formulas 2 2 2 1 cos2 sin 2 1 cos 2 cos 2 1 cos2 tan 1 cos 2 θ θ θ θ θ θ θ- = + =- = + Example 63 Write an expression for 4 cos θ that does not have powers on the trigonometric functions greater than 1. Example 64 Write an equivalent expression for sin 4 x that does not contain powers of trigonometric functions greater than 1. ( ) 4 2 2 2 1 cos2 1 cos2 sin sin sin 2 2 1 cos2 1 2cos2 1 2cos2 cos 2 2 4 4 3 3cos2 2 4cos2 1 cos2 8 8 x x x x x x x x x x x x-- = = + - + - + = -- + + = = Half-Angle Identities sin x 2 = ± 1 – cos x 2 cos x 2 = ± 1 + cos x 2 tan x 2 = ± 1 – cos x 1 + cos x = sin x 1 + cos x = 1 – cos x sin x where the sign is determined by the quadrant in which x 2 lies. Example 65 Find the exact value of cos112.5 ° Solution Because 112.5° = 225° / 2 , we use the half-angle formula for cos α / 2 with α = 225°. What sign should we use when we apply the formula? Because 112.5° lies in quadrant II, where only the sine and cosecant are positive, cos 112.5° < 0. Thus, we use the - sign in the half-angle formula. Example 66 Verify the identity: 1 cos2 tan sin 2 θ θ θ- = Half-Angle Formulas for: 1 cos tan 2 sin sin tan 2 1 cos α α α α α α- = = + Example 67 Verify the identity: tan csc cot 2 α α α =- Example 68 Verify the following identity: 2 (sin cos ) 1 sin 2 θ θ θ- = - Solution: 2 2 2 (sin cos ) sin 2sin cos cos 1 cos2 1 cos2 2sin cos 2 2 2 2sin cos 1 sin 2 2 θ θ θ θ θ θ θ θ θ θ θ θ θ- =- +- + = +- =- = - Section 7.5 Product-to-Sum and Sum-to-Product Formulas [ ] [ ] [ ] [ ] 1 sin sin cos( ) cos( ) 2 1 cos cos cos( ) cos( ) 2 1 sin cos sin( ) sin( ) 2 1 cos sin sin( ) sin( ) 2 α β α β α β α β α β α β α β α β α β α β α β α β =-- + =- + + = + +- = +-- Example 69 Express each of the following as a sum or a difference: a. sin8 sin3 x x b. sin 4 cos x x Example 70 Express the following product as a sum or difference: cos3 cos2 x x sin sin 2sin cos 2 2 sin sin 2sin cos...
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2 Example 60 If 5 sin 13 θ = and θ lies in quadrant II...

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