Question

If $\sin \alpha +\sin \beta =0=\cos \alpha +\cos \beta$, where $0<\beta <\alpha <2\pi$, then which of the following is correct?

• A. $\alpha =\pi -\beta$
• B. $\alpha =\pi +\beta$
• C. $\alpha =2\pi -\beta$
• D. $2\alpha =\pi +2\beta$

Answer 1

The correct option is A $\alpha =\pi -\beta$

$sin\alpha +sin\beta =0\Rightarrow sin\alpha =-sin\beta \Rightarrow \alpha =\pi -\beta or2\pi -\beta$

and also

$cos\alpha +cos\beta =0\Rightarrow cos\alpha =-cos\beta \Rightarrow \alpha =\pi -\beta or\pi +\beta$

Since, $\alpha =\pi -\beta$ is satisfying both equations

then the Answer is $A$