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Question Number 193548 by lmcp1203 last updated on 16/Jun/23

Commented by lmcp1203 last updated on 16/Jun/23

$$hintbytrigonometryx=10$$

Answered by Subhi last updated on 16/Jun/23

$$$$$$\frac{a}{sin\left(30\right)}=\frac{b}{sin\left(100\right)}\Rrightarrow a=\frac{b}{2sin\left(100\right)}\left(i\right)$$$$\frac{b}{sin\left(x\right)}=\frac{c}{sin\left(10\right)}\Rrightarrow c=\frac{sin\left(10\right)b}{sin\left(x\right)}\left(ii\right)$$$$\frac{a}{sin(20-x)}=\frac{c}{sin\left(20\right)}\Rrightarrow \frac{a}{c}=\frac{sin(20-x)}{sin\left(20\right)}=\frac{i}{ii}$$$$\frac{sin\left(x\right)}{2sin\left(100\right).sin\left(10\right)}=\frac{sin(20-x)}{sin\left(20\right)}=\frac{sin\left(20\right)cos\left(x\right)-sin\left(x\right)cos\left(20\right)}{sin\left(20\right)}$$$$2sin\left(20\right)sin\left(100\right)sin\left(10\right)cos\left(x\right)-2sin\left(100\right)sin\left(10\right)cos\left(20\right)sin\left(x\right)=sin\left(20\right)sin\left(x\right)$$$$(sin\left(20\right)+\left(2sin\left(100\right)sin\left(10\right)cos\left(20\right)\right)sin\left(x\right)=\left(2sin\left(20\right)sin\left(100\right)sin\left(10\right)\right)cos\left(x\right)$$$$x={tan}^{-1}\left(\frac{2sin\left(20\right)sin\left(100\right)sin\left(10\right)}{sin\left(20\right)+\left(2sin\left(100\right)sin\left(10\right)cos\left(20\right)\right)}\right)=10$$

Commented by Subhi last updated on 16/Jun/23

Commented by lmcp1203 last updated on 16/Jun/23

$$thanks.$$

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