Given that,$$f(n$$θ$$)=2sin$$⁡$$2$$θ$$cos$$⁡$$2$$θ−$$cos$$⁡$$3n$$θ,and $$f($$θ$$)+f(2$$θ$$)+f(3$$θ$$)+$$…$$+f(n$$θ$$)=$$$$sin$$⁡λθ$$sin$$⁡θ$$sin$$⁡μθ, then the value of μ−λ, is $$____$$.

Question

Given that,$$f(n$$θ$$)=2sin$$⁡$$2$$θ$$cos$$⁡$$2$$θ−$$cos$$⁡$$3n$$θ,and  $$f($$θ$$)+f(2$$θ$$)+f(3$$θ$$)+$$…$$+f(n$$θ$$)=$$$$sin$$⁡λθ$$sin$$⁡θ$$sin$$⁡μθ, then the value of μ−λ, is $$____$$.

Infinity Learn · Accepted Answer

Given that,$$f(n$$θ$$)=2sin$$⁡$$2$$θ$$cos$$⁡$$2$$θ−$$cos$$⁡$$4n$$θ$$=2sin$$⁡$$2$$θ$$2sin$$⁡(2n+1)θ$$sin$$⁡(2n$$−$$1)θ$$=$$$$sin$$⁡(2n+1)θ−(2n$$−$$1)θ$$sin$$⁡(2n+1)θ$$sin$$⁡(2n$$−$$1)θ$$=$$$$sin$$⁡(2n+1)θ$$cos$$⁡(2n$$−$$1)θ−$$cos$$⁡(2n+1)θ$$sin$$⁡(2n$$−$$1)θ$$sin$$⁡(2n+1)θ$$sin$$⁡(2n$$−$$1)θ$$=$$$$cot$$⁡(2n$$−$$1)θ−$$cot$$⁡(2n+1)θ⇒$$f($$θ$$)+f(2$$θ$$)+f(3$$θ$$)+$$…$$+f(n$$θ$$)=($$$$cot$$⁡θ−$$cot$$⁡$$3$$θ$$)+($$$$cot$$⁡$$3$$θ−$$cot$$⁡$$5$$θ$$)+($$$$cot$$⁡$$5$$θ−$$cot$$⁡$$7$$θ$$)=$$$$cot$$⁡θ−$$cot$$⁡(2n+1)θ$$=$$$$cos$$⁡θ$$sin$$⁡θ−$$cos$$⁡(2n+1)θ$$sin$$⁡(2n+1)θ$$=$$$$sin$$⁡$$2n$$θ$$sin$$⁡θ$$sin$$⁡(2n+1)θ⇒μ$$=2n+1$$⇒λ$$=2n$$⇒μ−λ$$=1$$

Multiple and sub- multiple Angles

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Given that, $f (nθ) = \frac{2 \sin 2 θ}{\cos 2 θ - \cos 3 nθ}$ ,and $f (θ) + f (2 θ) + f (3 θ) + \dots + f (nθ) = \frac{\sin λθ}{\sin θsin μθ}, then$ the value of $μ - λ, is$ ____.

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Multiple and sub- multiple Angles

Remember concepts with our Masterclasses.

Given that,f(nθ)=2sin⁡2θcos⁡2θ−cos⁡3nθ,and f(θ)+f(2θ)+f(3θ)+…+f(nθ)=sin⁡λθsin⁡θsin⁡μθ, then the value of μ−λ, is ____.

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Given that, $f (nθ) = \frac{2 \sin 2 θ}{\cos 2 θ - \cos 3 nθ}$ ,and $f (θ) + f (2 θ) + f (3 θ) + \dots + f (nθ) = \frac{\sin λθ}{\sin θsin μθ}, then$ the value of $μ - λ, is$ ____.