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Question Number 117046 by 1549442205PVT last updated on 09/Oct/20

$$\mathrm{Calculate}\mathrm{without}\mathrm{using}\mathrm{caculator}:$$$$\mathrm{a})-2\sqrt{2}\sin 10°(2\sin 35°-\frac{\sec 5°}{2}-\frac{\cos 40°}{\sin 5°})$$$$\mathrm{b})\sin 6°-\sin 42°-\sin 66°+\sin 78°$$

Answered by bemath last updated on 09/Oct/20

$$\left(\mathrm{a}\right)-4\sqrt{2}\sin 35°\sin 10°+\frac{\sqrt{2}\sin 10°}{\cos 5°}+\frac{2\sqrt{2}\cos 40°\sin 10°}{\sin 5°}=$$$$2\sqrt{2}(\cos 45°-\cos 25°)+2\sqrt{2}\sin 5°+4\sqrt{2}\cos 40°\cos 5°=$$$$2-2\sqrt{2}\cos 25°+2\sqrt{2}\sin 5°+2\sqrt{2}(\cos 45°+\cos 35°)$$$$4-2\sqrt{2}\cos 25°+2\sqrt{2}\sin 5°+2\sqrt{2}\cos 35°=$$$$4-2\sqrt{2}\cos 25°+2\sqrt{2}\cos 35°+2\sqrt{2}\sin 5°=$$$$4+2\sqrt{2}(\cos 35°-\cos 25°)+2\sqrt{2}\sin 5°=$$$$4+2\sqrt{2}(-2\sin 30°\sin 5°)+2\sqrt{2}\sin 5°$$$$4-2\sqrt{2}\sin 5°+2\sqrt{2}\sin 5°=4$$

Answered by bobhans last updated on 09/Oct/20

$$\left(\mathrm{b}\right)(\sin 78°-\sin 66°)-(\sin 42°-\sin 6°)=$$$$2\cos 72°\sin 6°-\left(2\cos 24°\sin 18°\right)=$$$$2\sin 18°\sin 6°-2\sin 66°\sin 18°=$$$$2\sin 18°(\sin 6°-\sin 66°)=$$$$2\sin 18°(-2\cos 36°\sin 30°)=$$$$-2\sin 18°\cos 36°=$$$$-2\left(\frac{\sqrt{5}-1}{4}\right)\left(\frac{\sqrt{5}+1}{4}\right)=-\frac{1}{2}$$$$\Rightarrow \sin 18°=\frac{\sqrt{5}-1}{4}$$$$\Rightarrow \cos 36°=\frac{\sqrt{5}+1}{4}$$$$$$

Answered by 1549442205PVT last updated on 09/Oct/20

$$\mathrm{Thank}\mathrm{Sirs}.$$$$\mathrm{Solution}:$$$$\mathrm{a})\mathrm{A}=-2\sqrt{2}(2\sin 10\sin 35-\frac{\sin 10}{2\cos 5}-\frac{\sin 10\cos 40}{\sin 5})$$$$=-2\sqrt{2}(2\sin 10\sin 35-\sin 5-2\cos 5\cos 40)$$$$-2\sqrt{2}[\cos 25-\cos 45-\sin 5-(\cos 45+\cos 35)]$$$$=-2\sqrt{2}[(\cos 25-\cos 35)-\sin 5-2\cos 45]$$$$=-2\sqrt{2}(2\sin 30\sin 5-\sin 5-\sqrt{2})$$$$=-2\sqrt{2}(2.\frac{1}{2}\sin 5-\sin 5-\sqrt{2})$$$$=-2\sqrt{2}(-\sqrt{2})=4$$$$\mathrm{b})\mathrm{B}=\sin 6°-\sin 42°-\sin 66°+\sin 78°$$$$=(\sin 6-\sin 66)+(\sin 78-\sin 42)$$$$=2\cos 36\sin (-30)+(2\cos 60\sin 18$$$$=-2.\frac{1}{2}\cos 36+2\frac{1}{2}\sin 18$$$$=\sin 18-\mathrm{co}36=\sin 18-\sin 54$$$$=-2\cos 36\sin 18=\frac{-2\cos 36\sin 18\cos 18}{\cos 18}$$$$=\frac{-\cos 36\sin 36}{\sin 72}=-\frac{1}{2}\frac{\sin 72}{\sin 72}=-\frac{1}{2}$$

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