If $$cos$$⁡α$$+2cos$$⁡β$$+3cos$$⁡γ$$=$$$$sin$$⁡α$$+2sin$$⁡β$$+3sin$$⁡γ$$=0$$ then the value of $$sin$$ $$3$$α$$+8sin$$⁡$$3$$β$$+27sin$$⁡$$3$$γ is

Question

If $$cos$$⁡α$$+2cos$$⁡β$$+3cos$$⁡γ$$=$$$$sin$$⁡α$$+2sin$$⁡β$$+3sin$$⁡γ$$=0$$ then the value of $$sin$$ $$3$$α$$+8sin$$⁡$$3$$β$$+27sin$$⁡$$3$$γ is

Infinity Learn · Accepted Answer

Let $$a=$$$$cos$$⁡α$$+isin$$⁡α$$b=$$$$cos$$⁡β$$+isin$$⁡β$$c=$$$$cos$$⁡γ$$+isin$$⁡γThen, $$a+2b+3c=($$$$cos$$⁡α$$+2cos$$⁡β$$+3cos$$⁡γ$$)$$                                        $$+i($$$$sin$$⁡α$$+2sin$$⁡β$$+3sin$$⁡γ$$)=0$$⇒$$a3+8b3+27c3=18abc$$⇒$$cos$$⁡$$3$$α$$+8cos$$⁡$$3$$β$$+27cos$$⁡$$3$$γ$$=18cos$$⁡$$($$α$$+$$β$$+$$γ$$)and$$ $$sin$$⁡$$3$$α$$+8sin$$⁡$$3$$β$$+27sin$$⁡$$3$$γ$$=18sin$$⁡$$($$α$$+$$β$$+$$γ$$)$$

If $\cos α + 2 \cos β + 3 \cos γ = \sin α + 2 \sin β + 3 \sin γ = 0$ then the value of sin $3 α + 8 \sin 3 β + 27 \sin 3 γ$ is

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If cos⁡α+2cos⁡β+3cos⁡γ=sin⁡α+2sin⁡β+3sin⁡γ=0 then the value of sin 3α+8sin⁡3β+27sin⁡3γ is

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If $\cos α + 2 \cos β + 3 \cos γ = \sin α + 2 \sin β + 3 \sin γ = 0$ then the value of sin $3 α + 8 \sin 3 β + 27 \sin 3 γ$ is