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Trigonometry calculator for right-angled triangles (SOHCAHTOA)

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Enter two things you know about a right-angled triangle (an angle and a side, or two sides), and the calculator finds everything else, saying which ratio it used:

sin⁡θ=opphypcos⁡θ=adjhyptan⁡θ=oppadj\sin\theta = \frac{\text{opp}}{\text{hyp}} \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}} \qquad \tan\theta = \frac{\text{opp}}{\text{adj}}

SOH CAH TOA is the usual way to remember them. Sides are named from the angle θ\theta: the hypotenuse is opposite the right angle (always the longest side), the opposite faces θ\theta, and the adjacent is next to θ\theta.

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Worked examples

  • Side from an angle: angle 30°, hypotenuse 10 cm. Opposite =10×sin⁡30°=5= 10 \times \sin 30° = 5 cm; adjacent =10×cos⁡30°≈8.66= 10 \times \cos 30° \approx 8.66 cm.
  • Angle from two sides: opposite 7 cm, adjacent 24 cm. tan⁡θ=7÷24\tan\theta = 7 \div 24, so θ=tan⁡−1(0.2917)≈16.26°\theta = \tan^{-1}(0.2917) \approx 16.26°. The hypotenuse is 25 cm.

How to choose the ratio

  1. Label the sides opp, adj and hyp from the angle you're using.
  2. Tick the two sides involved: the one you know and the one you want.
  3. Pick the ratio that uses those two: O and H → SOH, A and H → CAH, O and A → TOA.
  4. Substitute and rearrange. To find an angle, use the inverse function (sin⁡−1\sin^{-1}, cos⁡−1\cos^{-1} or tan⁡−1\tan^{-1}).

Exact values to know

0°30°45°60°90°
sin012\tfrac{1}{2}22\tfrac{\sqrt{2}}{2}32\tfrac{\sqrt{3}}{2}1
cos132\tfrac{\sqrt{3}}{2}22\tfrac{\sqrt{2}}{2}12\tfrac{1}{2}0
tan033\tfrac{\sqrt{3}}{3}13\sqrt{3}undefined

These come up on non-calculator papers: see our exact trig values questions.

Triangles without a right angle

SOH CAH TOA only works in right-angled triangles. For any other triangle, use the sine rule or the cosine rule. If you know two sides of a right-angled triangle and want the third, Pythagoras is quicker.

Common mistakes

  • Calculator in radians instead of degrees: check for a small D on the screen.
  • Labelling the sides from the wrong angle.
  • Dividing when you should multiply: if the unknown is on top of the fraction, multiply; if it's underneath, divide.

Practise with our trigonometry questions.

Exam tips

  • Write the ratio with the numbers substituted before rearranging.
  • Angles of elevation and depression are measured from the horizontal.
  • Round angles to 1 decimal place and lengths to 3 significant figures unless told otherwise.

Frequently asked questions

What does SOH CAH TOA mean?

Sin = Opposite ÷ Hypotenuse, Cos = Adjacent ÷ Hypotenuse, Tan = Opposite ÷ Adjacent.

How do you find an angle with trigonometry?

Find the ratio of the two sides you know, then use the inverse function, such as θ = tan⁻¹(opposite ÷ adjacent).

Which side is the hypotenuse?

The longest side, opposite the right angle.

Is anything I type sent to a server?

No. The calculation runs in your browser and nothing is stored.

Sources

  1. GCSE mathematics: subject content and assessment objectives. Department for Education.
  2. Cambridge IGCSE Mathematics 0580 syllabus. Cambridge International.

Keep reading

Trigonometry (SOH CAH TOA) questions and answers

8 GCSE and IGCSE questions with worked answers, checked as you type.

Pythagoras theorem calculator: find the missing side

The missing side of a right-angled triangle, with exact surds, decimals and the working.

Sine rule calculator with working

Missing sides and angles in any triangle with the sine rule, including the ambiguous case.

Pythagoras and right-triangle trigonometry

Use Pythagoras' theorem and sine, cosine and tangent to find sides and angles in right-angled triangles.