The function $f(x)=\sin3x - \sin2x + \sin x$ is defined for the domain $0 \le x \le \frac\pi2$.
a) By method of factorisation, show that $f(x) = \sin2x(2\cos x - 1)$.
b) Hence solve the equation $f(x) = 0$ using the given domain.
c) Using small angles identities, approximate the value of $f(0.1)$.
After a while I got through with part
a) show that $f(x) = \sin2x(2\cos x - 1)$
$\sin3x - \sin2x + \sin x$
$= \sin(2x + x) - \sin2x + \sin x$
$= \sin2x\cos{x} + \cos2x\sin x - \sin2x + \sin x$
$= \sin2x \cos x + (2\cos^2x -1)\sin x - \sin2x + \sin x$
$= \sin2x\cos x + 2\cos^2x\sin x - \sin x - \sin2x + \sin x$
$= \sin2x\cos x + 2\cos^2x\sin x - \sin2x$
$= \sin2x\cos x + 2\sin x\cos^2x - 2\sin x\cos x$
$= \sin2x\cos x + 2\sin x\cos x(\cos x - 1)$
$= \sin2x\cos x + \sin2x(\cos x - 1)$
$= \sin2x(\cos x + \cos x -1)$
$=\sin2x(2\cos x-1)$
For part b) Not sure if right
$=> \sin2x(2\cos x - 1) = 0$
$2\cos x - 1 = 0$
$2\cos x = 1$
$\cos x = \frac{1}{2}$
x = 60 degrees or $\frac{\pi}{3}$
$(a)$ Using Prosthaphaeresis Formulas,
$$\sin3x+\sin x=2\sin2x\cos x$$
$(b)$ $$\sin2x(1-2\cos x)=0\implies$$
either $\displaystyle\sin2x=0\implies2x=n\pi\iff x=\frac{n\pi}2$ where $n$ is any integer (Find proper value of $n$)
or $\displaystyle1-2\cos x=0\iff\cos x=\frac12=\cos\frac\pi3\implies x=2m\pi\pm\frac\pi3$ where $m$ is any integer (Find proper value of $m$)
or both are zero (show this is not possible)
$(c)$ Using Power Series for Sine, $$\sin y\approx y-\frac{y^3}{3!}$$ for small $y$