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Question Number 170371 by cortano1 last updated on 22/May/22

solve ((sin (x+18°))/(sin x)) = ((sin 18°)/(sin 12°))

solvesin(x+18°)sinx=sin18°sin12°

Answered by greougoury555 last updated on 22/May/22

sin 18°=2sin 12°sin 48° ((sin (x+18°))/(sin x)) = ((2sin 12° sin 48°)/(sin 12°)) sin (x+18°)=2sin x sin 48° sin (x+18°)=cos (x−48°)−cos (x+48°) cos (72°−x)+cos (x+48°)=cos (x−48°) 2cos 60° cos (12°−x)=cos (x−48°) ⇒cos (12°−x)=cos (x−48°) ⇒x−48°=±(12°−x)+2kπ ⇒ { ((x−48°=12°−x+2kπ⇒x=30°+kπ)),((x−48°=−12°+x +2kπ (have no solution))) :}

sin18°=2sin12°sin48°sin(x+18°)sinx=2sin12°sin48°sin12°sin(x+18°)=2sinxsin48°sin(x+18°)=cos(x−48°)−cos(x+48°)cos(72°−x)+cos(x+48°)=cos(x−48°)2cos60°cos(12°−x)=cos(x−48°)⇒cos(12°−x)=cos(x−48°)⇒x−48°=±(12°−x)+2kπ⇒{x−48°=12°−x+2kπ⇒x=30°+kπx−48°=−12°+x+2kπ(havenosolution)

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