IB Maths AA HL Ahl 3 11 Hl Further Trigonometric Equations Questions

Practise IB Mathematics HL 3.11 by applying further trigonometric equations methods to questions from the Question Bank.

Exam points

Solve complex trigonometric equations on finite intervals using reciprocal, inverse or compound-angle identities to reduce them to a solvable form.
List and verify every valid root in the stated domain, preserving quadrant signs, identity restrictions and any excluded values.

[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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IB Maths AA HL Ahl 3 11 Hl Further Trigonometric Equations Questions question 1