Express cos 5x in terms of sine multiples of x

Question

Mathematics

Dimples

7 September, 10:33

Express cos 5x in terms of sine multiples of x

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Answers (1)

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Ezekiel Schmitt · Answer 1

cos x = [ e^ (ix) - e^ (-ix) ] / 2i along with [ A - B ]^5 =

(A^5 - B^5) - 5 AB (A^3 - B^3) + 10 A²B² (A - B) ... A = e^ (ix), B = e^ (-ix)

[ cos x ] ^5 = (1/32i) { [ e^ (5ix) - e^ (-5x) ] - 5 [ (e^ (3ix) - e^ (-3ix) ] + 10 [ e^ (ix) - e^ (-ix) ] }

= (1 / 16) { cos 5x - 5 cos 3x + 10 cos x } ... much faster