SOH CAH TOA — Trigonometry Made Simple

Trigonometry lets you find missing sides and angles in right-angled triangles. It sounds scary, but once you know the three formulas, it's just substitution and calculator work. Let's break it down.

The Three Sides

Every right-angled triangle has three sides. You label them based on the angle you're working with (not the right angle — the other one):

Hypotenuse (H) — the longest side, always opposite the right angle
Opposite (O) — the side directly opposite your angle
Adjacent (A) — the side next to your angle (that isn't the hypotenuse)

Key Tip

Think of it like this: you're standing at the angle. The side you're looking directly at is the Opposite. The side next to your feet is the Adjacent. The long slanted side is always the Hypotenuse.

The Three Formulas: SOH CAH TOA

SOH CAH TOA is a memory trick. Each three-letter chunk tells you a formula:

SOH: Sin(θ) = Opposite ÷ Hypotenuse

CAH: Cos(θ) = Adjacent ÷ Hypotenuse

TOA: Tan(θ) = Opposite ÷ Adjacent

To remember it, some people say "Some Old Horses Can Always Hear Their Owners Approach" or "Silly Old Harry Caught A Herring Trawling Off America". Use whichever sticks!

How to Choose Which Formula

Look at what you have and what you want:

Have O and H? Use SOH (Sin)
Have A and H? Use CAH (Cos)
Have O and A? Use TOA (Tan)

Finding a Missing Side

Worked Example 1: Finding the Opposite

A ladder leans against a wall. The angle between the ground and the ladder is 35°. The ladder is 6m long (hypotenuse). How high up the wall does it reach?

We want the Opposite. We have the Hypotenuse (6m) and the angle (35°).
O and H → use SOH: Sin(35°) = O ÷ 6
Rearrange: O = 6 × Sin(35°)
Calculator: O = 6 × 0.5736 = 3.44m (2 d.p.)

Worked Example 2: Finding the Adjacent

A ramp makes a 20° angle with the ground. The ramp is 5m long. How far along the ground does it extend?

We want the Adjacent. We have the Hypotenuse (5m) and 20°.
A and H → use CAH: Cos(20°) = A ÷ 5
Rearrange: A = 5 × Cos(20°)
Calculator: A = 5 × 0.9397 = 4.70m (2 d.p.)

Worked Example 3: Finding the Hypotenuse

A tree casts a shadow 8m long. The angle from the tip of the shadow to the top of the tree is 50°. How tall is the tree?

We want the Opposite (height of tree). We have the Adjacent (8m) and 50°.
O and A → use TOA: Tan(50°) = O ÷ 8
Rearrange: O = 8 × Tan(50°)
Calculator: O = 8 × 1.1918 = 9.53m (2 d.p.)

Finding a Missing Angle

When you know two sides but need the angle, you use the inverse function on your calculator — that's the sin¹, cos¹, or tan¹ button (sometimes shown as arcsin, arccos, arctan).

Worked Example 4: Finding an Angle Using Sin

A right-angled triangle has an opposite side of 4cm and a hypotenuse of 9cm. Find the angle.

O and H → use SOH: Sin(θ) = 4 ÷ 9 = 0.4444
Use inverse: θ = sin¹(0.4444)
Calculator: θ = 26.4° (1 d.p.)

Worked Example 5: Finding an Angle Using Tan

A football pitch slopes up 2m over a distance of 50m. What angle is the slope?

O = 2m, A = 50m → use TOA: Tan(θ) = 2 ÷ 50 = 0.04
Use inverse: θ = tan¹(0.04)
Calculator: θ = 2.3° (1 d.p.)

Worked Example 6: Finding an Angle Using Cos

A zip wire is 30m long. The platform is 25m horizontally from the landing point. What angle does the wire make with the ground?

A = 25m, H = 30m → use CAH: Cos(θ) = 25 ÷ 30 = 0.8333
Use inverse: θ = cos¹(0.8333)
Calculator: θ = 33.6° (1 d.p.)

Common Mistakes

Wrong mode on your calculator — make sure it says DEG (degrees), not RAD or GRAD. This is the number one mistake in exams.
Mixing up Opposite and Adjacent — always label the sides first, from the angle you're using.
Forgetting to use inverse — if you're finding an angle, you need sin¹/cos¹/tan¹, not sin/cos/tan.

Practice Questions

Test Yourself

A right-angled triangle has an angle of 40° and a hypotenuse of 10cm. Find the opposite side.
A right-angled triangle has an angle of 55° and an adjacent side of 7cm. Find the opposite side.
A right-angled triangle has an opposite side of 5cm and a hypotenuse of 13cm. Find the angle.
A slide is 4m long and the top is 2.5m above the ground. What angle does the slide make with the ground?
A kite string is 20m long at an angle of 65° to the ground. How high is the kite?

Answers

10 × Sin(40°) = 6.43cm
7 × Tan(55°) = 10.00cm
sin¹(5/13) = 22.6°
sin¹(2.5/4) = 38.7°
20 × Sin(65°) = 18.13m