When do I use SOHCAHTOA and when do I use Pythagoras?

Both only work in right-angled triangles. Use Pythagoras when the question involves three sides and no angles. Use SOHCAHTOA when an angle (other than the right angle) is involved, either given or asked for.

Why does my calculator give strange answers?

It is probably in radians mode instead of degrees. Look for a small D or DEG on the screen. If you see R or RAD, change the mode before you start. A quick test is sin 30, which should give 0.5.

What does sin⁻¹ mean?

It is the inverse sine. It works backwards from a ratio to an angle. If sin θ = 0.5, then θ = sin⁻¹(0.5) = 30°. It does not mean 1 divided by sin.

Maths explained

Trigonometry basics, SOHCAHTOA explained step by step

Learn trigonometry basics with SOHCAHTOA: label a right-angled triangle, find a missing side or angle with sin, cos and tan, and avoid the common traps.

For students. By the Qlive team. 12 October 2026 . 5 min read.

This guide is for you if you are starting trigonometry and want the basics to make sense. In CBSE it begins in Class 10 with Introduction to Trigonometry and Some Applications of Trigonometry (heights and distances). ICSE covers it in Classes 9 and 10, and it is a big part of GCSE and IGCSE maths. Everything here is about right-angled triangles, so it helps to know Pythagoras theorem first.

The idea in plain words

In a right-angled triangle, if you know one angle (other than the right angle), the ratios between the sides are always the same, no matter how big the triangle is. Trigonometry gives those ratios names: sine, cosine and tangent.

Step 1: label the sides

Pick the angle you are working with, usually called θ (theta). Then label:

Hypotenuse (H): the longest side, opposite the right angle
Opposite (O): the side directly across from θ
Adjacent (A): the side next to θ that is not the hypotenuse

The hypotenuse never changes. The opposite and adjacent swap if you switch to the other angle.

Step 2: remember SOHCAHTOA

Letters	Ratio	Meaning
SOH	sin θ = O/H	sine is opposite over hypotenuse
CAH	cos θ = A/H	cosine is adjacent over hypotenuse
TOA	tan θ = O/A	tangent is opposite over adjacent

Step 3: choose the ratio

Look at which two sides are involved: the one you know and the one you want. Pick the ratio that uses both.

Worked examples

Make sure your calculator is in degrees before you start.

Example 1: finding the opposite side

A right-angled triangle has an angle of 30° and a hypotenuse of 12 cm. Find the side opposite the 30° angle.

Known: H = 12. Wanted: O. The ratio with O and H is sin.
sin 30° = x/12
Multiply both sides by 12: x = 12 × sin 30°
sin 30° = 0.5, so x = 12 × 0.5 = 6

Answer: 6 cm

Example 2: using tan

The angle is 40° and the side adjacent to it is 10 cm. Find the opposite side.

Known: A = 10. Wanted: O. The ratio with O and A is tan.
tan 40° = x/10
x = 10 × tan 40° = 10 × 0.8391 = 8.391

Answer: 8.39 cm (to 2 decimal places)

Example 3: when the unknown is on the bottom

The angle is 35° and the opposite side is 7 cm. Find the hypotenuse.

Known: O = 7. Wanted: H. The ratio is sin.
sin 35° = 7/h
Multiply both sides by h: h × sin 35° = 7
Divide by sin 35°: h = 7 ÷ sin 35° = 7 ÷ 0.5736 = 12.20

Answer: 12.2 cm (to 1 decimal place)

Sense check: the hypotenuse is longer than 7, as it must be.

Example 4: finding an angle

A right-angled triangle has an opposite side of 5 cm and an adjacent side of 12 cm. Find the angle θ.

Known: O = 5 and A = 12. The ratio is tan.
tan θ = 5/12 = 0.4167
Use the inverse: θ = tan⁻¹(0.4167) = 22.6°

Answer: θ = 22.6° (to 1 decimal place)

Example 5: height of a tower (exact values)

From a point 20 m from the foot of a tower, the angle of elevation of the top is 60°. Find the height of the tower.

Draw it: the ground is the adjacent side (20 m), the tower is the opposite side (h), and the angle is 60°.
tan 60° = h/20
h = 20 × tan 60° = 20√3
Using √3 ≈ 1.732: h ≈ 34.64

Answer: 20√3 m, which is about 34.64 m

Exact values worth learning

CBSE and many exam boards expect you to know these without a calculator.

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	not defined

A pattern helps: sin goes 0, 1/2, 1/√2, √3/2, 1 as the angle rises, and cos is the same list in reverse.

Common mistakes and how to avoid them

Calculator in radians. Check that sin 30 gives 0.5 before every test.

Labelling from the wrong angle. Opposite and adjacent depend on which angle you are using. Mark θ clearly, then label.

Multiplying when you should divide. If the unknown is on the bottom of the fraction, as in Example 3, you divide. Rearrange step by step instead of guessing.

Using SOHCAHTOA on a triangle with no right angle. These ratios only work in right-angled triangles. Other triangles need the sine and cosine rules.

Rounding too early. Keep the full calculator value until the final step.

Reading sin⁻¹ as 1/sin. The inverse function gives an angle. It is a separate button, usually SHIFT then sin.

Try these

The hypotenuse is 20 cm and the angle is 30°. Find the opposite side.
The adjacent side is 8 cm and the angle is 45°. Find the opposite side.
The hypotenuse is 15 cm and the angle is 50°. Find the adjacent side, to 2 decimal places.
The opposite side is 9 cm and the hypotenuse is 15 cm. Find the angle, to 1 decimal place.
From 15 m away, the angle of elevation of the top of a pole is 60°. Find the height of the pole, to 2 decimal places.

Answers

O = 20 × sin 30° = 20 × 0.5 = 10 cm.
O = 8 × tan 45° = 8 × 1 = 8 cm.
A = 15 × cos 50° = 15 × 0.6428 = 9.64 cm.
sin θ = 9/15 = 0.6, so θ = sin⁻¹(0.6) = 36.9°.
h = 15 × tan 60° = 15√3 ≈ 15 × 1.732 = 25.98 m.

Still unsure?

Trigonometry mixes several older skills: rearranging equations, decimals and reading diagrams. If it still feels shaky, a Qlive teacher can work through it with you one-to-one, starting with a free assessment that finds the exact gap. You can read more on our maths page and our CBSE page.