Question

If $\cos (A-B)=\frac{1}{2}$, and $\sin (A+B)=\frac{1}{2}$, find the smallest positive values of $A$ and $B$ and also their most general values.

Answer 1

$cos(a-b)=\frac{1}{2}\Rightarrow a-b=2n\pi +\frac{z}{3}$

$sin(a+b)=\frac{1}{2}\Rightarrow a+b=2nz+\frac{z}{6}$

$a=2n\pi +\frac{1}{2}(\frac{\pi }{3}+\frac{z}{6})$

$a=2nz+\frac{\pi }{4}$

$b=2n\pi +\frac{1}{2}(\frac{\pi }{6}-\frac{z}{3})=2nz-\frac{z}{6}$

Smallest value of $n=0$

$a=\frac{\pi }{4}$ & $b=-\frac{\pi }{4}$ Answer.