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Question Number 94522 by Abdulrahman last updated on 19/May/20

$$\mathrm{if}\mathrm{sinA}+\mathrm{sinB}=\mathrm{n}\mathrm{and}\mathrm{cosA}+\mathrm{cosB}=\mathrm{m}$$$$\mathrm{then}\sin (\mathrm{A}+\mathrm{B})=?$$$$\mathrm{a}:\frac{3\mathrm{mn}}{{\mathrm{m}}^2+{\mathrm{n}}^2}\mathrm{b}:\frac{2\mathrm{mn}}{{\mathrm{m}}^2+{\mathrm{n}}^2}\mathrm{c}:\frac{\mathrm{mn}}{{\mathrm{m}}^2+{\mathrm{n}}^2}\mathrm{d}:\frac{2\mathrm{mn}}{\mathrm{m}+\mathrm{n}}$$$$\mathrm{with}\mathrm{steps}?$$

Answered by i jagooll last updated on 19/May/20

$$\mathrm{squaring}\mathrm{eq}\left(1\right)\mathrm{\&}\left(2\right)\mathrm{then}\mathrm{adding}$$

Commented by Abdulrahman last updated on 19/May/20

$$\mathrm{please}\mathrm{solve}\mathrm{it}$$

Answered by john santu last updated on 19/May/20

$$\cos (\mathrm{A}-\mathrm{B})=\frac{{\mathrm{m}}^2+{\mathrm{n}}^2-2}{2}$$$$\sin (\mathrm{A}+\mathrm{B})=\frac{2\mathrm{mn}-[\sin 2\mathrm{A}+\sin 2\mathrm{B}]}{2}$$$$2\sin (\mathrm{A}+\mathrm{B})=2\mathrm{mn}-[2\sin (\mathrm{A}+\mathrm{B}).\cos (\mathrm{A}-\mathrm{B})]$$$$2\sin (\mathrm{A}+\mathrm{B})=2\mathrm{mn}-\left[\frac{{\mathrm{m}}^2+{\mathrm{n}}^2-2}{2}\times 2\sin (\mathrm{A}+\mathrm{B})\right]$$$$[2+{\mathrm{m}}^2+{\mathrm{n}}^2-2]\sin (\mathrm{A}+\mathrm{B})=2\mathrm{mn}$$$$\sin (\mathrm{A}+\mathrm{B})=\frac{2\mathrm{mn}}{{\mathrm{m}}^2+{\mathrm{n}}^2}$$

Commented by Abdulrahman last updated on 19/May/20

$$\mathrm{bundle}\mathrm{of}\mathrm{thanks}$$

Answered by som(math1967) last updated on 19/May/20

$$\mathrm{sinA}+\mathrm{sinB}=\mathrm{n}$$$$2\sin \frac{\mathrm{A}+\mathrm{B}}{2}\cos \frac{\mathrm{A}-\mathrm{B}}{2}=\mathrm{n}$$$$\mathrm{cosA}+\mathrm{cosB}=\mathrm{m}$$$$2\cos \frac{\mathrm{A}+\mathrm{B}}{2}\cos \frac{\mathrm{A}-\mathrm{B}}{2}=\mathrm{m}$$$$\therefore \tan \frac{\mathrm{A}+\mathrm{B}}{2}=\frac{\mathrm{n}}{\mathrm{m}}$$$$\sin \frac{\mathrm{A}+\mathrm{B}}{2}=\frac{\mathrm{n}}{\sqrt{{\mathrm{m}}^2+{\mathrm{n}}^2}}$$$$\cos \frac{\mathrm{A}+\mathrm{B}}{2}=\frac{\mathrm{m}}{\sqrt{{\mathrm{m}}^2+{\mathrm{n}}^2}}$$$$\sin (\mathrm{A}+\mathrm{B})$$$$2\sin \frac{\mathrm{A}+\mathrm{B}}{2}\cos \frac{\mathrm{A}+\mathrm{B}}{2}$$$$=\frac{2\mathrm{mn}}{{\mathrm{m}}^2+{\mathrm{n}}^2}$$$$\mathrm{so}\mathrm{b})\frac{2\mathrm{mn}}{{\mathrm{m}}^2+{\mathrm{n}}^2}\mathrm{is}\mathrm{ans}$$

Commented by peter frank last updated on 19/May/20

$$thankyou$$

Commented by Abdulrahman last updated on 19/May/20

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