Solution assignment 08 Unit circle and simple formulas

Show:

$\sin(3x)=3\sin(x)-4\sin^3(x)$

To show this relation we have to use a formula which is less often used (see Addition and Subtraction formulas):

$\sin(u+v)=\sin(u)\cos(v)+\sin(v)\cos(v)$

Using this formula we can write the function $\sin(3x)$ as follows:

$\sin(3x)=\sin(2x+x)=\sin(2x)\cos(x)+\sin(x)\cos(2x)$

In the summary formulas are given for $\sin(2x)$ and $\cos(2x)$ which can better be known by heart. We mean the following formulas:

$\sin(2x)\cos(x)+\sin(x)\cos(2x)=$

$=2\sin(x)\cos^2(x)+\sin(x)(1-2\sin^2(x))=$

$=2\sin(x)(1-\sin^2(x))+\sin(x)-2\sin^3(x)=$

$=2\sin(x)-2\sin^3(x)+\sin(x)-2\sin^3(x)=$

$=3\sin(x)-4\sin^3(x)$