Key ideas
SOH CAH TOA links an angle in a right-angled triangle to two of its sides.
sin x° = OH
cos x° = AH
tan x° = OA
The selection process
- Label the sides from the angle: Hypotenuse (longest, opposite the right angle), Opposite (across from the angle), Adjacent (next to the angle).
- Tick the two sides involved: the one you know and the one you want (or, for an angle, the two you know).
- Choose: write out SOH CAH TOA and tick those two letters wherever they appear. The group with two ticks is your ratio. For example, knowing O and wanting H: SOHCAHTOA → sin.
- Substitute the numbers, then solve.
⚠ Your calculator must be in degrees (a small D or DEG on the screen). In radians you'll get strange answers.
Labelling the sides
Label the sides of the triangle as hypotenuse, opposite and adjacent, from the marked angle.(1 mark)
Find the hypotenuse first.p is opposite the right angle → p = hypotenuse (H)
In plain EnglishThe hypotenuse is always the longest side, and it's always across from the right angle. It doesn't depend on which angle is marked.
Find the opposite side.q is across from the marked angle → q = opposite (O)
In plain EnglishStand at the marked angle and look straight across the triangle. The side you see is the opposite.
The one left is the adjacent.r → adjacent (A)✓1
In plain EnglishThe adjacent side touches the marked angle, but it isn't the hypotenuse.
⚠ Watch out. If you mark a different angle, the opposite and adjacent swap over. The hypotenuse never changes.
Choosing the ratio
Which ratio should you use to find x?(1 mark)
Label the sides.15 cm is the hypotenuse (H). x is opposite the 32° angle (O).
In plain EnglishLabel from the 32° angle.
Tick what you know and what you want.Know: H Want: O
In plain EnglishWe know the hypotenuse and we want the opposite. The adjacent isn't involved, so cross it out.
Find the pair in SOH CAH TOA.SOHCAHTOAO and H → use sin✓1
In plain EnglishWrite out SOH CAH TOA and tick O and H wherever they appear. The only group with both ticked is SOH, so use sin x° = O over H.
Finding a side
Calculate the length of side x. Give your answer to 1 decimal place.(3 marks)
Label, tick and choose.Know H = 12, want OSOHCAHTOA → sin
In plain EnglishFrom the 35° angle: 12 cm is the hypotenuse, x is opposite. O and H together means sin.
Substitute into the formula.sin x° = OHsin 35° = x12✓1
In plain EnglishPut the angle in, and the two sides into the fraction. x is on top.
Rearrange: multiply.x = 12 × sin 35°✓2
In plain Englishx is being divided by 12, so multiply both sides by 12 to get x on its own.
Calculate and round.x = 6.8829…x = 6.9 cm✓3
In plain EnglishType 12 × sin(35) into your calculator in degree mode. Round to 1 decimal place and include the units.
Calculate the length of side x. Give your answer to 1 decimal place.(3 marks)
Label, tick and choose.Know A = 8, want OSOHCAHTOA → tan
In plain EnglishFrom the 52° angle: 8 m is next to the angle (adjacent), x is across from it (opposite). O and A together means tan.
Substitute.tan 52° = x8✓1
In plain EnglishOpposite on top, adjacent on the bottom.
Multiply and calculate.x = 8 × tan 52°✓2x = 10.2 m✓3
In plain English8 × tan(52) = 10.2395… which rounds to 10.2 m.
When the unknown is on the bottom
Calculate the length of side x. Give your answer to 1 decimal place.(3 marks)
Label, tick and choose.Know A = 9, want HSOHCAHTOA → cos
In plain EnglishFrom the 40° angle: 9 cm is adjacent and x is the hypotenuse. A and H together means cos.
Substitute.cos 40° = 9x✓1
In plain EnglishThis time x is on the bottom of the fraction.
Rearrange: x and cos 40° swap places.x = 9cos 40°✓2
In plain EnglishWhen the unknown is on the bottom, x and cos 40° swap places: x = 9 over cos 40°, which means divide. Check: it should come out bigger than 9, because the hypotenuse is the longest side.
Calculate and round.x = 11.7487…x = 11.7 cm✓3
In plain EnglishOn your calculator, 9 ÷ cos 40 = 11.7487… which rounds to 11.7 cm. ✓ Bigger than 9.
⚠ Common mistake. Multiplying (9 × cos 40° = 6.9) gives an answer shorter than 9. The hypotenuse can't be shorter than another side, so that must be wrong.
Finding an angle
Calculate the size of angle x°. Give your answer to 1 decimal place.(3 marks)
Label, tick and choose.Know O = 7 and A = 10SOHCAHTOA → tan
In plain EnglishFrom angle x: 7 cm is opposite and 10 cm is adjacent. O and A together means tan.
Substitute.tan x° = 710✓1
In plain EnglishThis time both sides are known and the angle is missing.
Use the inverse.x = tan⁻¹(710)✓2
In plain EnglishTo undo tan, use tan⁻¹ (press SHIFT then tan on most calculators).
Calculate and round.x = 34.9920…x = 35.0°✓3
In plain EnglishOn your calculator, tan⁻¹(7 ÷ 10) = 34.9920… which rounds to 35.0°.
Problems in context
A 6 m ladder leans against a wall. It makes an angle of 70° with the ground. How far up the wall does the ladder reach? Give your answer to 1 decimal place.(3 marks)
Sketch it as a right-angled triangle.ladder = hypotenuse = 6 m; height up the wall = opposite
In plain EnglishThe wall and the ground meet at a right angle. The ladder is the slanted side, so it's the hypotenuse. The 70° is at the bottom of the ladder.
Choose and substitute.SOHCAHTOA → sinsin 70° = h6✓1
In plain EnglishKnow H, want O. Tick O and H in SOH CAH TOA: SOH, so sin.
Solve.h = 6 × sin 70°✓2h = 5.6 m✓3
In plain English6 × sin(70) = 5.6382… which rounds to 5.6 m.
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Your turn
Calculator needed, in degree mode. Give lengths and angles to 1 decimal place. Stuck? Tap Need a hint?
Past paper questions
Ready for exam-style questions? These collect Intermediate 1 and Standard Grade General past paper questions, with answers.
SOH CAH TOA: calculating a lengthPast paper questions where you find a missing side.Dynamic Maths Open ↗ (opens in a new tab) SOH CAH TOA: calculating an anglePast paper questions where you find a missing angle.Dynamic Maths Open ↗ (opens in a new tab)