詳解：馬克勞林級數展開式為

\(f(x) \buildrel\textstyle.\over= f(0) + \frac{{f'(0)}}{{1!}}x + \frac{{f''(0)}}{{2!}}{x^2} + \frac{{f'''(0)}}{{3!}}{x^3} + \frac{{{f^{(4)}}(0)}}{{4!}}{x^4} +  \cdots \)

\(f(x) = \sin x\) \(  \Rightarrow f(0) = \sin (0) = 0\)

\(f'(x) = \cos x\) \(  \Rightarrow f'(0) = \cos (0) = 1\)

\(f''(x) =  - \sin x\) \(  \Rightarrow f''(0) =  - \sin (0) = 0\)

\(f'''(x) =  - \cos x\) \(  \Rightarrow f'''(0) =  - \cos (0) =  - 1\)

\({f^{(4)}}(x) = \sin x\) \(  \Rightarrow {f^{(4)}}(0) = \sin (0) = 0\)

\(\sin x \buildrel\textstyle.\over= 0 + \frac{1}{{1!}}x + \frac{0}{{2!}}{x^2} + \frac{{ - 1}}{{3!}}{x^3} + \frac{0}{{4!}}{x^4} +  \cdots \)=

\(\frac{1}{{1!}}x - \frac{1}{{3!}}{x^3} + \frac{1}{{5!}}{x^5} - \frac{1}{{7!}}{x^7} +  \cdots \)