Trigonometry Lecture Notes_part2

# Θ θ θ θ θ θ example 62 verify the identity 3

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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 cos2 cos 2 1 cos2 tan 1 cos2 θ θ θ θ θ θ θ - = + = - = + Example 63

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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

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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 2 2 cos cos 2cos cos 2 2 cos cos 2sin sin 2 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 71 Express each sum or difference as a product: a. sin9 sin5 x x + b. cos4 cos3 x x - Example 72

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Express the difference as a product: sin 4 sin 2 x x - Example 73 Verify the following identity: sin sin tan cot sin sin 2 2 x y x y x y x y + + - = - Example 74 Verify the following identity: cos3 cos5 tan sin3 sin5 x x x x x - = + Section 7.2 Trigonometric Equations This section involves equations that have a trigonometric expression with a variable, such as cos x . To understand this section we must consider a simple equation such as cos x = 0.5.
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