IB Maths AA HL 3.11 Further trigonometric equations Question Bank

Practise IB Mathematics HL 3.11 by applying further trigonometric equations methods to exam-style questions.

Exam points

identify the mathematical structure, variable or representation
select and apply the correct theorem, formula or algorithm
check the result using units, domain, graph or logical reasoning

[Maximum number: 6]

Hence or otherwise solve the equation $\sin x+\sin 3 x=\cos x$ in the interval $0<x<\pi$ .

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 $x=\frac{5 \pi}{12}$

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AHL 3.11 (HL)—Further trigonometric equations question 1