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Question Number 84117 by Power last updated on 09/Mar/20

Answered by TANMAY PANACEA last updated on 09/Mar/20

$$sin100+sin20+sin50$$$$2sin60.cos40+cos40$$$$cos40(1+\sqrt{3})$$$$96\times 2\times sin40.cos40+\frac{1}{2}(cos30-cos100)$$$$=96\times 2\times sin40.cos40+\frac{\sqrt{3}}{4}-\frac{cos100}{2}$$$$=192sin40.cos40+\frac{\sqrt{3}}{4}-\frac{1}{2}(2{cos}^{2}50-1)$$$$=192sin40.cos40-{sin}^{2}40+\frac{\sqrt{3}+2}{4}$$$$$$$$wait$$$$$$

Answered by mind is power last updated on 09/Mar/20

$$ithinkQuationis$$$$\frac{96sin\left(80\right)sin\left(65\right)sin\left(35\right)}{sin\left(20\right)+sin\left(50\right)+sin\left(110\right)}$$$$sin\left(65\right)sin\left(35\right)=\frac{1}{2}(cos(65-35)-cos(65+35))$$$$=\frac{1}{2}(cos\left(30\right)-cos\left(110\right))$$$$sin\left(80\right).sin\left(65\right)sin\left(35\right)=\frac{sin\left(80\right)}{2}(cos\left(30\right)-cos\left(110\right))$$$$=\frac{1}{2}(sin\left(80\right)cos\left(30\right)-sin\left(80\right)cos\left(110\right))$$$$=\frac{1}{4}(sin\left(110\right)+sin\left(50\right)-sin\left(180\right)-sin(-20))$$$$=\frac{1}{4}(sin\left(110\right)+sin\left(50\right)+sin\left(20\right))$$$$\iff$$$$\frac{sin\left(65\right)sin\left(35\right)sin\left(80\right)}{sin\left(110\right)+sin\left(50\right)+sin\left(20\right)}=\frac{1}{4}$$$$\iff \frac{96sin\left(65\right)sin\left(35\right)sin\left(80\right)}{sin\left(110\right)+sin\left(20\right)+sin\left(50\right)}=\frac{96}{4}=24$$$$$$$$$$

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