# Expand sin^6 x?

Mar 9, 2018

${\sin}^{6} \left(x\right) = - \frac{1}{32} \left(\cos \left(6 x\right) - 6 \cos \left(4 x\right) + 15 \cos \left(2 x\right) - 10\right)$

#### Explanation:

We want to expand

${\sin}^{6} \left(x\right)$

One way is to use these identities repeatedly

• ${\sin}^{2} \left(x\right) = \frac{1}{2} \left(1 - \cos \left(2 x\right)\right)$
• ${\cos}^{2} \left(x\right) = \frac{1}{2} \left(1 + \cos \left(2 x\right)\right)$

This often gets quite long, which sometimes leads to mistake

Another way is to use the complex numbers (and Euler's formula)

We can express sine and cosine as

$\textcolor{red}{\sin \left(x\right) = \frac{{e}^{i x} - {e}^{- i x}}{2 i}}$ and $\textcolor{red}{\cos \left(x\right) = \frac{{e}^{i x} + {e}^{- i x}}{2}}$

Thus

${\sin}^{6} \left(x\right) = {\left(\frac{{e}^{i x} - {e}^{- i x}}{2 i}\right)}^{6}$

$= {\left({e}^{i x} - {e}^{- i x}\right)}^{6} / \left(- 64\right)$

$= \frac{{e}^{6 i x} - 6 {e}^{4 i x} + 15 {e}^{2 i x} - 20 + 15 {e}^{- 2 i x} - 6 {e}^{- 4 i x} + {e}^{- 6 i x}}{- 64}$

$= - \frac{1}{32} \frac{{e}^{6 i x} + {e}^{- 6 i x} - 6 {e}^{4 i x} - 6 {e}^{- 4 i x} + 15 {e}^{2 i x} + 15 {e}^{- 2 i x} - 20}{2}$

$= - \frac{1}{32} \left(\cos \left(6 x\right) - 6 \cos \left(4 x\right) + 15 \cos \left(2 x\right) - 10\right)$

The third way is using De Moivre's theorem, we can express

$\textcolor{b l u e}{2 \cos \left(n x\right) = {z}^{n} + \frac{1}{z} ^ n}$ and color(blue)(2isin(nx)=z^n-1/z^n

where ${z}^{n} = {\left(\cos \left(x\right) + i \sin \left(x\right)\right)}^{n} = \cos \left(n x\right) + i \sin \left(n x\right)$

Thus

${\left(2 i \sin \left(x\right)\right)}^{6} = {\left(z - \frac{1}{z}\right)}^{6}$

$\implies {\sin}^{6} \left(x\right) = - \frac{1}{64} {\left(z - \frac{1}{z}\right)}^{6}$

Expand the binomial on the right hand side

$B I N = \left({z}^{6} + \frac{1}{z} ^ 6 - 6 {z}^{4} - \frac{6}{z} ^ 4 + 15 {z}^{2} + \frac{15}{z} ^ 2 - 20\right)$

$\textcolor{w h i t e}{R H S} = \left({z}^{6} + \frac{1}{z} ^ 6 - 6 \cdot \left({z}^{4} + \frac{1}{z} ^ 4\right) + 15 \cdot \left(z + \frac{1}{z}\right) + 20\right)$

$\textcolor{w h i t e}{R H S} = \left(2 \cos \left(6 x\right) - 12 \cos \left(4 x\right) + 30 \cos \left(2 x\right) - 20\right)$

Thus

${\sin}^{6} \left(x\right) = - \frac{1}{64} \left(2 \cos \left(6 x\right) - 12 \cos \left(4 x\right) + 30 \cos \left(2 x\right) - 20\right)$

${\sin}^{6} \left(x\right) = - \frac{1}{32} \left(\cos \left(6 x\right) - 6 \cos \left(4 x\right) + 15 \cos \left(2 x\right) - 10\right)$