Question

If $\alpha ,\beta$ be two angle belonged to the interval $[0,4\pi ]$ such that $\sin \alpha +\cos \beta =2$ then the number of possible values of the pair $(\alpha ,\beta )$ is

• A. $5$
• B. $2$
• C. $4$
• D. $6$

Answer 1

The correct option is D $6$

$\sin \alpha +\cos \beta =2$

solution is only possible if,
$\sin \alpha =1$ and $\cos \beta =1$

so in $\alpha ,\beta \in [0,4\pi ]$, solutions are,

$\alpha =\frac{\pi }{2},\frac{5\pi }{2}$ and $\beta =0,2\pi ,4\pi$

hence possible number of ordered pair $(\alpha ,\beta )$ is 6.