Let n be an odd integer. If sin ntheta = suml | vedclass

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Question

(b) Given $\sin \,n	heta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}	heta } $

==> $\sin n	heta = {b_0}{\sin ^0}	heta + {b_1}\sin {\,^1}	heta

Vedclass · Accepted Answer

${b_0} = 0,{b_1} = n$

Vedclass · Answer

(b) Given $\sin \,n	heta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}	heta } $

==> $\sin n	heta = {b_0}{\sin ^0}	heta + {b_1}\sin {\,^1}	heta $

$ + {b_2}{\sin ^2}	heta + {b_3}{\sin ^3}	heta + ..... + {b_n}{\sin ^n}	heta $

==> $\sin n	heta = {b_0} + {b_1}\sin 	heta + {b_2}{\sin ^2}	heta + .... + {b_n}{\sin ^n}	heta $

($n$ is an odd integer)

$ = {\,^n}{C_1}\sin 	heta .{(1 - {\sin ^2}	heta )^{(n - 1)/2}}$

$ - {\,^n}{C_3}{\sin ^3}	heta {(1 - {\sin ^2}	heta )^{(n - 3)/2}} + ....$

$	herefore \,\,\,{b_0} = 0,{b_1} = $ coefficient of $\sin 	heta = {\,^n}{C_1} = n$

($ n -1= n -3$ are all even integers)

Let $n$ be an odd integer. If $\sin n\theta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}\theta } $ for every value of $\theta $, then

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