Question

Assertion :The equation $\sin x=1$, has infinite number of solutions Reason: The domain of $f\left(x\right)=\sin x$ is $(-\infty ,\infty )$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

The general solution of $\sin \theta =\sin \alpha$ is given by $\theta =m\pi +(-1{)}^m\alpha$

Now, if m is an even integer i.e., $m=2n$ (where n ∈ Z) then,

$\theta =2n\pi +\frac{\pi }{2}$

$\Rightarrow \theta =(4n+1)\frac{\pi }{2}$

Again, if $m$ is an odd integer i.e. $m=2n+1$ (where n ∈ Z) then,

$\theta =(2n+1)\cdot \pi -\frac{\pi }{2}$

$\Rightarrow \theta =(4n+1)\frac{\pi }{2}.$

Hence, the general solution of $\sin \theta =1$ is $\theta =(4n+1)\frac{\pi }{2},n\in Z$

$\sin \left(x\right)=1$ has only one solution if $x=[0,2\pi ]$.

However in the above equation the domain is not specified.

Since $x$ can take the value of any real number, hence $\sin x=1$ has infinite solutions.