STEP trigonometry: guide and practice problems
STEP trigonometry tests whether you can move fluently between identities, equations and geometry.
Key ideas
- Multiple-angle formulae (sin 3θ, cos 3θ, tan 3θ) turn special angles into roots of polynomials.
- Sum-to-product formulae factorise sums of sines and cosines, so equations split into simple pieces.
- In triangle problems, the largest angle is opposite the longest side; the cosine rule decides which.
- Area = (1/2)ab sin C and inradius = area / semi-perimeter connect lengths and angles.
Common traps
- Dividing by a trig function that can be zero and losing solutions.
- Giving solutions outside the stated interval.
- Using degrees and radians in the same calculation.
Practice problems
Original problems in the style of STEP. Try each one for at least 20 minutes before you open a hint; write a full solution, then compare.
Problem 1: A cubic in tan
- Show that \(\tan3\theta=\dfrac{3t-t^3}{1-3t^2}\), where \(t=\tan\theta\).
- Show that the roots of \(t^3-3t^2-3t+1=0\) are \(\tan\tfrac{\pi}{12}\), \(\tan\tfrac{5\pi}{12}\) and \(\tan\tfrac{3\pi}{4}\).
- Hence find the exact value of \(\tan^2\tfrac{\pi}{12}+\tan^2\tfrac{5\pi}{12}\).
Hint 1
\(\tan3\theta=\tan(2\theta+\theta)\).
Hint 2
The cubic says \(3t-t^3=1-3t^2\), that is \(\tan3\theta=1\). One root is \(-1\); factorise.
Full solution
(i) \(\tan2\theta=\frac{2t}{1-t^2}\), and \(\tan(2\theta+\theta)=\dfrac{\frac{2t}{1-t^2}+t}{1-\frac{2t^2}{1-t^2}}=\dfrac{3t-t^3}{1-3t^2}\).
(ii) The cubic rearranges to \(3t-t^3=1-3t^2\), so (where \(1-3t^2\ne0\)) \(\tan3\theta=1\): \(3\theta=\tfrac\pi4,\tfrac{5\pi}4,\tfrac{9\pi}4\), so \(\theta=\tfrac\pi{12},\tfrac{5\pi}{12},\tfrac{3\pi}4\), giving three different values of \(t\).
(iii) \(t^3-3t^2-3t+1=(t+1)(t^2-4t+1)\), so \(\tan\frac\pi{12}\) and \(\tan\frac{5\pi}{12}\) are the roots of \(t^2-4t+1=0\): sum \(4\), product \(1\). The sum of squares is \(4^2-2=14\).
Results: \(14\).
2 more trigonometry problems
A triangle with sides in arithmetic progression; Three sines. Each with two hints and a full solution.
Next steps
Related topics: Complex numbers and polynomials · Mechanics. Stuck? Read the problem-solving strategies. Ready for the real thing? Official STEP past papers.
STEP is run by OCR. A Level Math Revision is independent: it is not affiliated with or endorsed by OCR, the University of Cambridge or any university. Every problem here is our own, written in the style of STEP; for real papers use OCR's official past papers.