An identity without losing solutions

Choose your method, connect the parts and explain your conclusions. These original questions include written reasoning and sketches as well as exact answer checks.

Suggested marks guide how much working to show. The site does not automatically award examination marks for proofs, diagrams or methods. Use scaffolded fluency practice when you need a hint first.

A-level prerequisite. Course filters use selected objective mappings; none of these tasks establishes complete paper coverage.

Before you start

Trigonometric equations in intervals. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

Solve $\sin(2x)=\cos x$ , for $0\leq x<2\pi$ .

Part a

Show how to reduce the equation without discarding any solutions.

Part b

Give all solutions in increasing order, as coefficients of π.

Part c

Explain why solving only the sine equation is incomplete.

Model solution and review criteria

Part a

Use sin(2x)=2sin x cos x and rearrange to cos x(2sin x − (1))=0. Dividing immediately by cos x would lose its zero solutions.

Use the double-angle identity.
Factor rather than divide by a possibly zero factor.

Part b

cos x=0 gives $\frac{\pi}{2}$ and $\frac{3\pi}{2}$ . sin x= $\frac{1}{2}$ gives $\frac{\pi}{6}$ and $\frac{5\pi}{6}$ . All four lie in the stated half-open interval.

Solve both factors.
Use both relevant quadrants.
Check the interval and completeness.

Part c

The product is zero when either factor is zero. The original equation is satisfied at the two cosine zeros even though they do not satisfy the resulting sine equation.

Identify the lost branch.
Check it in the original equation.

Compare your own reasoning. No automatic examination marks are awarded.