Question

Rewrite with only sin x and cos x. sin 3x

Answer 1

$\bf sin({{ \alpha}} + {{ \beta}})=sin({{ \alpha}})cos({{ \beta}}) + cos({{ \alpha}})sin({{ \beta}}) \\\\\\ sin(2\theta)=2sin(\theta)cos(\theta) \\ \quad \\ cos(2\theta)= \begin{cases} \boxed{cos^2(\theta)-sin^2(\theta)}\\ 1-2sin^2(\theta)\\ 2cos^2(\theta)-1 \end{cases}\\\\ -------------------------------\\\\ sin(3x)\implies sin(2x+x)\implies sin(2x)cos(x)+cos(2x)sin(x)$

$\bf 2sin(x)cos(x)cos(x)~+~[cos^2(x)-sin^2(x)]sin(x) \\\\\\ \stackrel{like~terms}{\stackrel{\downarrow }{2sin(x)cos^2(x)}~~+~~\stackrel{\downarrow }{sin(x)cos^2(x)}}-sin^3(x) \\\\\\ 3sin(x)cos^2(x)-sin^3(x)$