Question

What is $\sin \left(x+\frac{\mathrm{\pi }}{2}\right)$?

Answer 1

Find the value of $\mathbf{sin}\mathbf{}\left(x+\frac{\pi }{2}\right)$:

As we know,

$\sin (A+B)=\sin A\cos B+\sin B\cos A$

In the given question, $A=x$ and $B=\frac{\mathrm{\pi }}{2}$

So,

$$\begin{gathered} \sin \left(x+\frac{\mathrm{\pi }}{2}\right)=\sin x·\cos \frac{\mathrm{\pi }}{2}+\sin \frac{\mathrm{\pi }}{2}·\cos x \\ =\sin x·0+1·\cos x\left[\because \cos \frac{\mathrm{\pi }}{2}=0\mathrm{and}\sin \frac{\mathrm{\pi }}{2}=1\right] \\ =\cos x \end{gathered}$$

Hence, $\mathbf{sin}\mathbf{}\mathbf{(}\mathbf{x}\mathbf{+}\frac{\mathbf{\pi }}{\mathbf{2}}\mathbf{)}\mathbf{=}\mathbf{cos}\mathbf{}\mathit{x}$