Question

If $\sin \left(A+B\right)=\cos (A-B)=\frac{\sqrt{3}}{2}andA,B,\left(A>B\right)$ are acute angles, find the value of $A$ and $B$

Answer 1

Given, $\sin \left(A+B\right)=\cos (A-B)=\frac{\sqrt{3}}{2}$

or, $\sin (A+B)=\frac{\sqrt{3}}{2}$

or, $A+B={60}^o$......(1). [ Since $A,B$ are acute]

Also given,

$\cos (A-B)=\frac{\sqrt{3}}{2}$

or, $A-B={30}^o$.......(2). [ Since $A,B$ are acute]

Solving (1) and (2) we get,

$A={45}^o,B={15}^o$.