If 0 A B , sin A + sin B = 3 2 and cos A + cos B = 1 2 , then A is equal to

Question

If $0 < A < B < π,$ $\sin A + \sin B = \sqrt{\frac{3}{2}}$ and $\cos A + \cos B = \frac{1}{\sqrt{2}},$ then $A$ is equal to

Smart Achievers · Accepted Answer

(1)

Clearly, ${(\sin A + \sin B)}^{2} + {(\cos A + \cos B)}^{2} = 2$

$∴$ $2 + 2 (\sin A \sin B + \cos A \cos B) = 2$

$\Rightarrow \cos (B - A) = 0 \Rightarrow B = A + 90 °$

Second equation gives,

$\cos A - \sin A = \frac{1}{\sqrt{2}}$

$\Rightarrow \cos (A + 45 °) = \cos 60 °$

$∴$ $A = 15 °$

Want Us to Email you About Special Offers & Updates?