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Question Number 70052 by ahmadshahhimat775@gmail.com last updated on 30/Sep/19

$$$$$$\sin A+\sin B=nand\cos A+\cos B=m$$$$\sin (A+B)=?$$

Answered by $@ty@m123 last updated on 30/Sep/19

$$\sin A+\sin B=n$$$$\Rightarrow 2\sin \frac{A+B}{2}\cos \frac{A-B}{2}=n....\left(1\right)$$$$\cos A+\cos B=m$$$$\Rightarrow 2\cos \frac{A+B}{2}\cos \frac{A-B}{2}=m...\left(2\right)$$$$Dividing\left(1\right)by\left(2\right),weget$$$$\tan \frac{A+B}{2}=\frac{n}{m}$$$$Wehave,$$$$\sin (A+B)=\frac{2\tan \frac{A+B}{2}}{1+\tan {}^2\frac{A+B}{2}}$$$$=\frac{\frac{2n}{m}}{1+\frac{n^2}{m^2}}=\frac{2mn}{m^2+n^2}$$

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