Question

If $0\le x,y\le {180}^o$ and $\sin (x-y)=\cos (x+y)=\frac{1}{2}$, then the values of $x$ and $y$ are given by

• A. $x={45}^o,y={15}^o$
• B. $x={45}^o,y={135}^o$
• C. $x={165}^o,y={15}^o$
• D. $x={165}^o,y={135}^o$

Answer 1

Answer: (A), (D)

The correct options are
A $x={45}^o,y={15}^o$
D $x={165}^o,y={135}^o$

Given $\sin (x-y)=\frac{1}{2}$
$\Rightarrow x-y={30}^o$ or ${150}^o$ ($\because 0\le x,y\le {180}^o$) ....(1)
Also given $\cos (x+y)=\frac{1}{2}$
$\Rightarrow x+y={60}^o$ and ${300}^o$ ($\because 0\le x,y\le {180}^o$) ....(2)
So, from (1) and (2) we get
When $x={45}^o,y={15}^o$, or $x={165}^o,y={135}^o$.