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Question Number 212445 by ChantalYah last updated on 13/Oct/24

Commented by Frix last updated on 13/Oct/24

$$80\cos A?150\sin A=13$$$$$$$$80\cos A-150\sin A=13$$$$-170\sin (A-{\tan }^{-1}\frac{8}{15})=13$$$$\sin (A-{\tan }^{-1}\frac{8}{15})=-\frac{13}{170}$$$$A=\{\begin{array}{l}2n\pi +{\tan }^{-1}\frac{8}{15}-{\sin }^{-1}\frac{13}{170}\\ (2n+1)\pi +{\tan }^{-1}\frac{8}{15}+{\sin }^{-1}\frac{13}{170}\end{array}$$$$$$$$80\cos A+150\sin A=13$$$$170\sin (A+{\tan }^{-1}\frac{8}{15})=13$$$$\sin (A+{\tan }^{-1}\frac{8}{15})=\frac{13}{170}$$$$A=\{\begin{array}{l}2n\pi -{\tan }^{-1}\frac{8}{15}+{\sin }^{-1}\frac{13}{170}\\ (2n+1)\pi -{\tan }^{-1}\frac{8}{15}-{\sin }^{-1}\frac{13}{170}\end{array}$$

Answered by mr W last updated on 13/Oct/24

$$\frac{8}{\sqrt{8^2+{15}^2}}\cos A-\frac{15}{\sqrt{8^2+{15}^2}}\sin A=\frac{13}{10\sqrt{8^2+{15}^2}}$$$$\cos \alpha \cos A-\sin \alpha \sin A=\frac{13}{170}$$$$\cos (\alpha +A)=\frac{13}{170}$$$$\Rightarrow \alpha +A=2k\pi \pm {\cos }^{-1}\frac{13}{170}$$$$\Rightarrow A=2k\pi \pm {\cos }^{-1}\frac{13}{170}-{\cos }^{-1}\frac{8}{17}$$

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