There exist constants b , and d such that ( sin &#x2061;<!-- ⁡ --> x ) 7

mistergoneo7 2022-07-15 Answered
There exist constants b, and d such that
( sin x ) 7 = a sin 7 x + b sin 5 x + c sin 3 x + d sin x
for all angles x. Find d
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Answers (2)

thatuglygirlyu
Answered 2022-07-16 Author has 14 answers
Evaluate the equation for different x:
π 6 a + b + 2 c + d = 1 64 π 4 a b +   c + d = 8 64 π 3 + a b + d = 27 64 π 2 a + b   c + d = 64 64
(after normalization of the coefficient of d)
Elimination by the combination ( 1 ) + 3 × ( 3 ) + 2 × ( 4 ) yields
6 d = 210 64 .
The evaluation for x = π 4 was not even necessary.

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Jorden Pace
Answered 2022-07-17 Author has 4 answers
sin 7 x = sin 6 x sin x = 1 16 16 sin 6 x sin x = 1 16 ( 4 sin 3 x ) 2 sin x = 1 16 ( 3 sin x sin 3 x ) 2 sin x = 1 16 ( 9 sin 2 x 6 sin x sin 3 x + sin 2 3 x ) sin x = 1 16 [ 9 2 ( 1 cos 2 x ) 3 ( cos 2 x cos 4 x ) + 1 2 ( 1 cos 6 x ) ] sin x = 9 32 ( sin x sin x cos 2 x ) 3 16 ( sin x cos 2 x sin x cos 4 x ) + 1 32 ( sin x sin x cos 6 x ) = 9 32 [ sin x 1 2 ( sin 3 x sin x ) ] 3 32 [ ( sin 3 x sin x ) ( sin 5 x sin 3 x ) ] + 1 32 [ sin x 1 2 ( sin 7 x sin 5 x ) ]
The coefficient of sin x in the above expression is
9 32 + 9 64 + 3 32 + 1 32 = 35 64

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