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Question Number 183254 by HeferH last updated on 24/Dec/22

Answered by TUN last updated on 24/Dec/22

$$\frac{sin\left(14x\right)}{sin\left(8x\right)}=\frac{sin\left(x\right)}{sin\left(7x\right)}$$

Answered by a.lgnaoui last updated on 25/Dec/22

$$△\mathrm{ABC}14\mathrm{x}+\theta =\pi \theta =\pi -14\mathrm{x}$$$$△\mathrm{ABD}6\mathrm{x}+\alpha +\theta =\pi \Rightarrow \alpha =8\mathrm{x}$$$$△\mathrm{ADC}8\mathrm{x}+\beta =\pi \beta =\pi -8\mathrm{x}$$$$\mathrm{AB}=\mathrm{CD}=\mathrm{a};\mathrm{AC}=\mathrm{BD}=\mathrm{b}$$$$1\bullet △\mathrm{ABD}$$$$\frac{\sin 6\mathrm{x}}{\mathrm{b}}=\frac{\sin 14\mathrm{x}}{\mathrm{AD}}=\frac{\sin 8\mathrm{x}}{\mathrm{a}}$$$$\mathrm{a}\times \sin 6\mathrm{x}=\mathrm{bsin}8\mathrm{x}$$$$\frac{a}{b}=\frac{\sin 8x}{\sin 6x}$$$${\mathrm{AD}}^2=a^2+b^2+2ab\times \cos 14x\left(1\right)$$$$$$$$2\bullet △\mathrm{BCD}$$$$\frac{\sin 7\mathrm{x}}{\mathrm{a}}=\frac{\sin 8\mathrm{x}}{\mathrm{b}}=\frac{\sin \mathrm{x}}{\mathrm{AD}}$$$$\mathrm{a}\times \sin 8\mathrm{x}=\mathrm{b}\times \sin 7\mathrm{x}$$$$\frac{\mathrm{a}}{\mathrm{b}}=\frac{\sin 7\mathrm{x}}{\sin 8\mathrm{x}}$$$${\mathrm{AD}}^2={\mathrm{a}}^2+{\mathrm{b}}^2-2\mathrm{ab}\times \cos \mathrm{x}\left(2\right)$$$$$$$$3\bullet △\mathrm{ABC}$$$$\frac{\sin 13\mathrm{x}}{\mathrm{a}+\mathrm{b}}=\frac{\sin \mathrm{x}}{\mathrm{a}}=\frac{\sin 14\mathrm{x}}{\mathrm{b}}\left(3\right)$$$$\sin 13\mathrm{x}=\sin 14\mathrm{x}+\sin \mathrm{x}\left(3\right)$$$$$$$$\left(1\right)-\left(2\right)$$$$2\mathrm{ab}(\cos 14\mathrm{x}+\cos \mathrm{x})=0$$$$\cos 14\mathrm{x}=\cos (\pi -x)$$$$14\mathrm{x}=\pi -x+k\times 2\pi$$$$\mathrm{x}=\frac{(2\mathrm{k}+1)\pi }{15}$$$$k=0\mathrm{x}=\frac{\pi }{15}k=4\mathrm{x}=\frac{3\pi }{5}$$$$k=1\mathrm{x}=\frac{\pi }{5}k=5\mathrm{x}=\frac{11\pi }{15}$$$$k=2\mathrm{x}=\frac{\pi }{3}k=6\mathrm{x}=\frac{13\pi }{5}$$$$k=3\mathrm{x}=\frac{7\pi }{15}$$$$\bullet cas:\mathrm{x}=\frac{\pi }{15}6\mathrm{x}=\frac{6\pi }{15}7\mathrm{x}=\frac{6\pi }{15}$$$$\alpha =\frac{8\pi }{15}\beta =\frac{7\pi }{15}\theta =\frac{\pi }{15}$$$$6x+\theta +\alpha =\frac{6\pi +\pi +8\pi }{15}=\pi (△ABD)$$$$8x+\beta =\frac{8\pi +7\pi }{15}=\pi (△BCD)$$$$\left(Asuivrepourautrescas\right)\Rightarrow$$

Commented by a.lgnaoui last updated on 25/Dec/22

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