IB Maths AA HL Ahl 3 11 Hl Further Trigonometric Equations Topic Practice

EITHER
$\sin x+\sin 3 x=\cos x \Rightarrow \frac{1-\cos 4 x}{2 \sin x}=\cos x$$\Rightarrow 1-\cos 4 x=2 \sin x \cos x,(\sin x \neq 0)$$\Rightarrow 1-\left(1-2 \sin ^{2} 2 x\right)=\sin 2 x$$\Rightarrow \sin 2 x(2 \sin 2 x-1)=0$$\Rightarrow \sin 2 x=0$ or $\sin 2 x=\frac{1}{2}$$2 x=\pi, 2 x=\frac{\pi}{6}$ and $2 x=\frac{5 \pi}{6}$

OR
$\sin x+\sin 3 x=\cos x \Rightarrow 2 \sin 2 x \cos x=\cos x$$\Rightarrow(2 \sin 2 x-1) \cos x=0,(\sin x \neq 0)$$\Rightarrow \sin 2 x=\frac{1}{2}$ or $\cos x=0$$2 x=\frac{\pi}{6}, 2 x=\frac{5 \pi}{6}$ and $x=\frac{\pi}{2}$

THEN
$\therefore x=\frac{\pi}{2}, x=\frac{\pi}{12}$ and $x=\frac{5 \pi}{12}$