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Question Number 43728 by Cheyboy last updated on 14/Sep/18

Commented by A.Haq.Soomro last updated on 15/Sep/18

$$\text{You can't use 'macro parameter character \#' in math mode}$$

Answered by MrW3 last updated on 16/Sep/18

$$letAB=1$$$$$$$$\frac{AE}{\sin 80°}=\frac{AB}{\sin 30°}$$$$\Rightarrow AE=\frac{\sin 80°}{\sin 30°}\times AB=2\sin 80°=2\cos 10°$$$$$$$$\frac{BD}{\sin 80°}=\frac{AB}{\sin 40°}$$$$\Rightarrow BD=\frac{\sin 80°}{\sin 40°}\times AB=2\cos 40°$$$$$$$$\frac{BE}{\sin 70°}=\frac{AB}{\sin 30°}$$$$\Rightarrow BE=\frac{\sin 70°}{\sin 30°}\times AB=2\sin 70°=2\cos 20°$$$$$$$${DE}^2={BD}^2+{BE}^2-2\times BD\times BE\times \cos 20°$$$$=4{\cos }^{2}40°+4{\cos }^{2}20°-8\cos 40°\cos 20°\cos {20}^2$$$$=4({\cos }^{2}40°+{\cos }^{2}20°-2\cos 40°{\cos }^{2}20°)$$$$=4[{(2{\cos }^{2}20°-1)}^2+{\cos }^{2}20°-2(2\times {\cos }^{2}20°-1){\cos }^{2}20°]$$$$=4[4{\cos }^{4}20°-4{\cos }^{2}20°+1+{\cos }^{2}20°-4{\cos }^{4}20°+2{\cos }^{2}20°]$$$$=4(1-{\cos }^{2}20°)=4{\sin }^{2}20°$$$$\Rightarrow DE=2\sin 20°$$$$$$$$\frac{CE}{\sin 10°}=\frac{AE}{\sin 20°}$$$$\Rightarrow CE=\frac{\sin 10°}{\sin 20°}\times AE=\frac{\sin 10°}{\sin 20°}\times 2\cos 10°=\frac{\sin 20°}{\sin 20°}=1$$$$$$$$\frac{CE}{\sin \mathrm{\angle }CDE}=\frac{DE}{\sin 20°}$$$$\Rightarrow \sin \mathrm{\angle }CDE=\frac{CE}{DE}\times \sin 20°=\frac{1\times \sin 20°}{2\sin 20°}=\frac{1}{2}$$$$\Rightarrow \mathrm{\angle }CDE=30°$$$$$$$$\Rightarrow x=\mathrm{\angle }CDE-10°=30°-10°=20°$$$$$$$$Forthisresultone{doesn}^{\prime }tneedtouse$$$$acalculator.$$

Commented by peter frank last updated on 22/Apr/20

$$thankyou$$

Commented by peter frank last updated on 22/Apr/20

$$thankyou$$

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