The sine function 𝑠𝑖𝑛(𝑥) can be expanded into a Taylor series:

• $sin(x)=\frac{x^1}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}...$

This is also a power series expansion of the sine function

Proof

We start with the definition of a Taylor series

• $f(x)=\frac{f(a)(x-a{)}^0}{0!}+\frac{f^{\prime }(a)(x-a{)}^1}{1!}+\frac{f^{\prime \prime }(a)(x-a{)}^2}{2!}+\frac{f^{\prime \prime \prime }(a)(x-a{)}^3}{3!}+\frac{f^{\prime \prime \prime \prime }(a)(x-a{)}^4}{4!}+...$
• $f(x)=\frac{f(0)x^0}{0!}+\frac{f^{\prime }(0)x^1}{1!}+\frac{f^{\prime \prime }(0)x^2}{2!}+\frac{f^{\prime \prime \prime }(0)x^3}{3!}+\frac{f^{\prime \prime \prime \prime }(0)x^4}{4!}+...\text{ let }a\text{ = 0}$
• $sin(x)=\frac{sin(0)x^0}{0!}+\frac{si{n}^{\prime }(0)x^1}{1!}+\frac{si{n}^{\prime \prime }(0)x^2}{2!}+\frac{si{n}^{\prime \prime \prime }(0)x^3}{3!}+\frac{si{n}^{\prime \prime \prime \prime }(0)x^4}{4!}+...\text{ let }f(x)=sin(x)$
• $sin(x)=\frac{sin(0)x^0}{0!}+\frac{cos(0)x^1}{1!}+\frac{-sin(0)x^2}{2!}+\frac{-cos(0)x^3}{3!}+\frac{sin(0)x^4}{4!}+...\text{ evaluate derivatives}$
• $sin(x)=\frac{0x^0}{0!}+\frac{1x^1}{1!}+\frac{0x^2}{2!}+\frac{-1x^3}{3!}+\frac{0x^4}{4!}+...$
• $sin(x)=0+\frac{x^1}{1!}+0-\frac{x^3}{3!}+0+...$
• $sin(x)=\frac{x^1}{1!}-\frac{x^3}{3!}+...$