Question

What is the formula of ${\sin }^3\left(x\right)$​?

Answer 1

Determine the formula for ${\sin }^3\left(x\right)$.

Use the trigonometric identity, $\mathit{s}\mathit{i}\mathit{n}\left(3x\right)\mathbf{=}\mathbf{3}\mathit{s}\mathit{i}\mathit{n}\left(x\right)\mathbf{-}\mathbf{4}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{3}}\left(x\right)$ and simplify as follow:

$\begin{array}{rcl}\sin \left(3x\right)-3\sin \left(x\right)& =& -4{\sin }^3\left(x\right)\\ & \Rightarrow & {\sin }^3\left(x\right)=-\frac{1}{4}\left[\sin \left(3x\right)-3\sin \left(x\right)\right]\\ & \Rightarrow & 4{\sin }^3\left(x\right)=-\frac{1}{4}\sin \left(3x\right)+\frac{3}{4}\sin \left(x\right)\\ & \Rightarrow & {\sin }^3\left(x\right)=\frac{3}{4}\sin \left(x\right)-\frac{1}{4}\sin \left(3x\right)\end{array}$

Hence, the formula is $\begin{array}{rcl}{\sin }^3\left(x\right)& =& \frac{3}{4}\sin \left(x\right)-\frac{1}{4}\sin \left(3x\right)\end{array}$.