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Trigonometry practice: SOHCAHTOA questions

SOHCAHTOA gives the three trigonometry ratios in a right-angled triangle: sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent. Label the sides from the marked angle, pick the ratio using the two sides you know and want, then solve.

SOHCAHTOA is the memory aid for the three trigonometric ratios in a right-angled triangle: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.

Year 9GCSE FoundationGCSE Higher

The method, step by step

  1. Label the triangle. Hypotenuse opposite the right angle, opposite across from the marked angle, adjacent next to it.
  2. Note which two sides matter. One you know, one you want.
  3. Pick the ratio. Whichever of SOH, CAH or TOA uses exactly those two sides.
  4. Substitute and solve. Rearrange for the unknown before reaching for the calculator.

Worked example

A right-angled triangle has a 30° angle with an opposite side of 7cm. Find the hypotenuse.
Opposite and hypotenuse are involved, so use SOH: sin 30° = opposite ÷ hypotenuse.
sin 30° = 0.5 exactly, so 0.5 = 7 ÷ hypotenuse.
Rearrange: hypotenuse = 7 ÷ 0.5.
The hypotenuse is 14cm

Common mistakes

Labelling opposite and adjacent from the right angle rather than from the marked angle.

Leaving the calculator in radians. For GCSE it must be in degrees.

SOHCAHTOA exam-style questions

Untimed practice with worked answers. Every answer is a whole number. This extra activity does not count towards your daily progress.

  1. Worked answer for question 1

    Opposite and hypotenuse, so use SOH: opp = 10 × sin 30° = 10 × 0.5 = 5 cm.

  2. Worked answer for question 2

    Opposite and adjacent, so use TOA: opp = 8 × tan 45° = 8 × 1 = 8 cm.

  3. Worked answer for question 3

    Adjacent and hypotenuse, so use CAH: adj = 20 × cos 60° = 20 × 0.5 = 10 cm.

  4. Worked answer for question 4

    sin x = 7/14 = 0.5, so x = sin⁻¹(0.5) = 30°.

  5. Worked answer for question 5

    opp = 25 × sin 35° = 25 × 0.5736 = 14.34, which is 14 cm to the nearest cm.

Continue with Year 9 daily practice

Common questions

What does SOHCAHTOA stand for?

Sin = Opposite ÷ Hypotenuse, Cos = Adjacent ÷ Hypotenuse, Tan = Opposite ÷ Adjacent. Label the sides from the angle you are using, then pick the ratio that uses the two sides you know or need.

When do you use inverse sin, cos or tan?

Use sin⁻¹, cos⁻¹ or tan⁻¹ when you know two sides and want the angle. Use sin, cos or tan when you know an angle and one side and want another side.

Are the SOHCAHTOA ratios on the GCSE formula sheet?

Yes. The three trigonometric ratios are on the GCSE Maths formulae sheet for both tiers, but the exact values for 0°, 30°, 45°, 60° and 90° are not, so learn those. See the GCSE Maths formula sheet.

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Printable worksheets for this topic

Free A4 PDFs with worked answers. See every printable, Year 1 to A-level.