GCSE Maths · Geometry & measures

Trigonometry

SOHCAHTOA in right-angled triangles, and the sine and cosine rules at Higher.

200 GCSE-style practice questions

Learn

Trigonometry, explained point by point

Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.

  1. Label the opposite, adjacent and hypotenuse, and choose the right ratio (SOH CAH TOA)

    The hypotenuse faces the right angle; the opposite faces the angle you are using; the adjacent is the remaining side. Then SOH CAH TOA picks the ratio from the two sides involved - opposite and hypotenuse means sin. Label the triangle before touching the calculator.

  2. Find a missing side in a right-angled triangle

    Choose the ratio, substitute, rearrange. To find the side opposite 35° when the hypotenuse is 12: sin 35° = x12, so x = 12 sin 35° ≈ 6.88. If the unknown is on the bottom of the fraction, you divide instead: x = 12 ÷ sin 35°.

  3. Find a missing angle in a right-angled triangle

    Form the ratio from the two known sides, then apply the INVERSE function: tan θ = 74 gives θ = tan⁻¹74 ≈ 60.3°. Make sure the calculator is in degrees - a wrong-mode answer looks plausible but is wrong.

  4. Know the exact values of sin, cos and tan for 0°, 30°, 45°, 60° and 90°

    These come up on the non-calculator paper: sin 30° = 12, cos 60° = 12, tan 45° = 1, sin 60° = 32, and friends. They all come from two triangles - half an equilateral triangle and a right-angled isosceles - so you can rebuild any value you forget.

  5. Use the sine rule, cosine rule and area = ½ab sin C in any triangleHigher only

    Without a right angle: the sine rule a/sin A = b/sin B links opposite pairs; the cosine rule a² = b² + c² − 2bc cos A handles two-sides-and-the-angle-between or all three sides. Area = ½ab sin C needs two sides and the included angle. Choose by what the question gives you.

  6. Apply trigonometry in 3D problemsHigher only

    Find a flat right-angled triangle inside the solid - often using a diagonal - and work one triangle at a time. The diagonal of a box needs Pythagoras across the base first, then again (or trig) up to the corner. Sketch each 2D triangle separately with its own labels.

Practice

Try a Trigonometry question

A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.

Work out the exact value of sin 60° × cos 30°.

  1. 12
  2. 34
  3. 32
  4. 34
Show the answer and the working

Answer: 34

sin 60° and cos 30° are both 32, so the product is 322 = 34.

  1. Recall sin 60° = 32 and cos 30° = 32.
  2. Multiply: 32 × 32 = 3×3 ÷ (2 × 2).
  3. 3 × 3 = 3 and 2 × 2 = 4.
  4. So the exact value is 34.

Make Trigonometry stick

A free account adds worked examples from our question bank, from a warm-up to a stretch, plus ask-Wiz - a coach you can put any point from this lesson to - and a checklist that remembers which points you've studied. No card needed.

Continue learning free

Ready to practise Trigonometry?

Start your free trial to unlock this topic - questions that adapt to how you're doing, with instant encouraging feedback on every answer. No card needed.

Start practising free

Already have an account? Log in

More Geometry & measures topics

Keep reading