Question

The most general value of $\theta$ satisfying the equations $\sin \theta =\sin \alpha$ and $\cos \theta =\cos \alpha$ is

• A. $2n\mathrm{\pi }+\mathrm{\alpha }$
• B. $2n\mathrm{\pi }-\mathrm{\alpha }$
• C. $n\mathrm{\pi }+\mathrm{\alpha }$

Answer 1

The correct option is C

$n\mathrm{\pi }+\mathrm{\alpha }$

Explanation for the correct option:

Find the general solution of the given equations

In the question, two equations are given,

$\sin \theta =\sin \alpha ...\left(1\right)$

And, $\cos \theta =\cos \alpha ...\left(2\right)$

Divide equation $\left(1\right)$ by equation $\left(2\right)$.

$$\begin{gathered} \frac{\sin \theta }{\cos \theta }=\frac{\sin \alpha }{\cos \alpha } \\ \Rightarrow \tan \theta =\tan \alpha \end{gathered}$$

We know that, $\tan x=\tan \left(n\mathrm{\pi }+\mathrm{x}\right)$.

So, $\tan \theta =\tan \left(n\mathrm{\pi }+\mathrm{\alpha }\right)$.

This implies that, $\theta =n\pi +\alpha$.

Hence, the most general value of $\theta$ satisfying the equations $\sin \theta =\sin \alpha$ and $\cos \theta =\cos \alpha$ is $\theta =n\pi +\alpha$.

Hence, option correct option is $C$,