Question

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathbf{\hspace{0.17em}}\mathit{x}\mathbf{,}\mathbf{\hspace{0.33em}}\mathit{a}\mathbf{=}\mathit{\pi }\mathbf{/}\mathbf{2}$

Short answer

$\sin \left(x\right)=1-\frac{{\left(x-\frac{\pi }{2}\right)}^2}{2}+\frac{x{\left(x-\frac{\pi }{2}\right)}^4}{24}-\frac{{\left(x-\frac{\pi }{2}\right)}^6}{720}+\dots$

Step-by-step solution

Given information.

The Maclaurin expansion of the function$f\left(x\right)=sin\hspace{0.17em}x,\hspace{0.33em}a=\pi /2$ is to be evaluated.

Definition of Maclaurin series.

The definition of the Maclaurin series is defined.

$\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{-}\frac{{\mathbf{x}}^{\mathbf{2}}}{\mathbf{2}}\mathbf{+}\frac{{\mathbf{x}}^{\mathbf{4}}}{\mathbf{24}}\mathbf{-}\frac{{\mathbf{x}}^{\mathbf{6}}}{\mathbf{720}}\mathbf{+}\mathbf{\dots }$

Begin by stating the Maclaurin series.

State the Maclaurin series.

$$\begin{gathered} \sin \left(x\right)=\sin \left[\frac{\pi }{2}+\left(x-\frac{\pi }{2}\right)\right] \\ =x-\frac{x^3}{6}+\frac{x^3}{120}+\dots \end{gathered}$$

Use the fact $\sin \left(x+\frac{\pi }{2}\right)=\cos \left(x\right)$to equate the previous equation.

$\sin \left[\frac{\pi }{2}+\left(x-\frac{\pi }{2}\right)\right]=\cos \left(x-\frac{\pi }{2}\right)$

$\cos \left(x\right)=1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+\dots$

Make appropriate substitution in the Maclaurin series and evaluate.

Replace x with the expression$\left(x-\frac{\pi }{2}\right)$in the third defined Maclaurin series.

$\sin \left(x\right)=\cos \left(x-\frac{\pi }{2}\right)$

localid="1657384732250" $\sin \left(x\right)=1-\frac{{\left(x-\frac{\pi }{2}\right)}^2}{2}+\frac{x{\left(x-\frac{\pi }{2}\right)}^4}{24}-\frac{{\left(x-\frac{\pi }{2}\right)}^6}{720}+\dots$

Thus, the final answer islocalid="1657384971196" $\sin \left(x\right)=1-\frac{{\left(x-\frac{\pi }{2}\right)}^2}{2}+\frac{x{\left(x-\frac{\pi }{2}\right)}^4}{24}-\frac{{\left(x-\frac{\pi }{2}\right)}^6}{720}+\dots$