Given f(α)=sin(2π+α)sin(−π−α)sin(π−α)cos(2π−α)sin(−α+23π). (1) Simplify f(α). (2) If α is an angle in the third quadrant and cos(α+3π)=53, find the value of f(α).
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Solution
(1) We have f(α)=sin(2π+α)sin(−π−α)sin(π−α)cos(2π−α)sin(−α+23π). Using the trigonometric identities, sin(π−α)=sinα, cos(2π−α)=cosα, sin(−α+23π)=−cosα, and sin(2π+α)=cosα, sin(−π−α)=sinα, the expression simplifies to f(α)=cosα⋅sinαsinα⋅cosα⋅(−cosα)=−cosα.
(2) Given that α is an angle in the third quadrant, and cos(α+3π)=53>0, so α+3π is an angle in the fourth quadrant. Thus, sin(α+3π)=−1−cos2(α+3π)=−1−(53)2=−54. Now, we simplify f(α): f(α)=−cosα=−cos[(α+3π)−3π]=−cos(α+3π)cos3π+sin(α+3π)sin3π=−53⋅21+(−54)⋅23=−103−1043=10−3−43.
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