# Maclaurin series of sine function

The sine function can be expressed as an infinite Maclaurin series:

${\displaystyle \sin {x}==x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots }$

This animation shows the first four terms of the series being added:

• the green curve shows the actual sine function
• the orange curves show each successive term
• the red curve shows the approximation as each new term is added

## Step by step

In this section we will add the first four terms of the Maclaurin series, one by one, and see how each new term improves the approximation.

This graph shows the sine function in green, and the first term of the expansion ${\displaystyle y=x}$ in red.

The orange curve below shows the second term ${\displaystyle -{\frac {x^{3}}{3!}}}$. The red curve show the effect of combining this with the previous term:

The orange curve below shows the third term ${\displaystyle {\frac {x^{5}}{5!}}}$. The red curve show the effect of combining this with the previous terms:

Finally, the orange curve below shows the fourth term ${\displaystyle -{\frac {x^{7}}{7!}}}$. The red curve show the effect of combining this with the previous terms: