First slide

Trigonometric transformations

Question

If $\cos A + \cos B = m$ and $\sin A + \sin B = n$ where $m, n \neq 0$ , then $\sin (A + B)$ is equal to

Moderate

A

$\frac{m n}{m^{2} + n^{2}}$
B

$\frac{2 m n}{m^{2} + n^{2}}$
C

$\frac{m^{2} + n^{2}}{2 m n}$
D

$\frac{m n}{m + n}$

Solution

We have,
$\begin{array}{l} m n = (\cos A + \cos B) (\sin A + \sin B) \\ \Rightarrow m n = \cos A \sin A + \sin (A + B) + \sin B \cos B \\ \Rightarrow 2 m n = \sin 2 A + \sin 2 B + \sin (A + B) \\ \Rightarrow 2 m n = 2 \sin (A + B) \cos (A - B) + \sin (A + B) . . . (i) \end{array}$
Also, we have
$m^{2} + n^{2} = 2 + 2 \cos (A - B)$ ... (ii)
From (i) and (ii), we have
$\sin (A + B) = \frac{2 m n}{m^{2} + n^{2}}$

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