Trigonometry Made Easy: 4 Worked Examples for Finding Missing Sides and Angles

Trigonometry is one of those topics that looks scary on paper but follows the same short routine every single time. Once you’ve got that routine, every right-angled triangle question on the paper works the same way: label, choose, substitute, solve.

This guide walks you through the method, then goes through four worked examples. Two find a missing side and two find a missing angle, which covers the two question types you’ll see in the exam.

What trigonometry actually does

Trigonometry is about the relationship between the sides and the angles of a right-angled triangle. The key idea is simple: for any given angle, the ratio between two particular sides is always the same, no matter how big or small the triangle is.

That means you can:

• use one angle and one side to find another side, or
• use two sides to find an unknown angle.

The angle we’re working with is usually labelled with the Greek letter θ (theta).

Step 1: Label the sides

Before anything else, label all three sides relative to the angle θ. This is the step students most often rush, and it’s where most marks are lost.

Side	Letter	How to spot it
Hypotenuse	H	The longest side, always opposite the right angle
Opposite	O	The side directly across from θ, not touching it
Adjacent	A	The side next to θ that isn’t the hypotenuse

Remember: The hypotenuse never changes, but the opposite and adjacent sides swap if you use the other acute angle. Always label from the angle given in the question.

Step 2: Choose your ratio

There are three trigonometric functions, one for each pair of sides:

Function	Ratio	Memory aid
sin θ	Opposite ÷ Hypotenuse	SOH
cos θ	Adjacent ÷ Hypotenuse	CAH
tan θ	Opposite ÷ Adjacent	TOA

Most students remember these as SOH CAH TOA. If you like a sillier memory aid, try: “Some Old Hags Can’t Always Hack Their Old Age.”

To pick the right function, look at the two sides that matter: the one you know and the one you’re trying to find. Whichever function uses both of them is the one you need.

Finding a missing side: Example 1

A right-angled triangle has an angle of 35°. The hypotenuse is 12 m. Find the side opposite the 35° angle, x.

1. Label: 12 m is the hypotenuse (H), and x is the opposite (O).
2. Choose: O and H means SOH, so we use sin.
3. Substitute: sin 35° = x ÷ 12
4. Solve: x = 12 × sin 35° = 6.88 m (3 s.f.)

The “friendly equation” trick

If you’re not sure whether to multiply or divide when you rearrange, write a simple equation you know is true next to your working:

3 = 6 ÷ 2

In Example 1, x is on the top of the fraction, in the same place as the 6. How do you get 6 from 3 and 2? You multiply them. So x = 12 × sin 35°.

Tip: Don’t round sin 35° to 0.57 and then multiply. 12 × 0.57 gives 6.84, but the correct answer is 6.88. Type the whole calculation in one go, 12 × sin(35), and round only at the end.

Finding a missing side: Example 2

A right-angled triangle has an angle of 48°. The side opposite the angle is 15 m. Find the adjacent side, x.

1. Label: 15 m is the opposite (O), and x is the adjacent (A).
2. Choose: O and A means TOA, so we use tan.
3. Substitute: tan 48° = 15 ÷ x
4. Solve: This time x is on the bottom of the fraction, where the 2 is in our friendly equation. How do you get 2 from 6 and 3? You divide: 2 = 6 ÷ 3. So:

x = 15 ÷ tan 48° = 13.5 m (3 s.f.)

Did you know? Sense-checking can catch errors instantly. tan 48° is about 1.11, which is bigger than 1, so the opposite side must be longer than the adjacent. Our answer (13.5 m) is shorter than 15 m, which is exactly what we’d expect. If you’d got 166.6 m by multiplying instead, the sense-check would tell you something had gone wrong.

Finding a missing angle

When you know two sides and want the angle, the first two steps stay the same. The only change is at the end, where you use an inverse trig function (sin⁻¹, cos⁻¹ or tan⁻¹) to go from the ratio back to the angle. On most scientific calculators, you press SHIFT (or 2nd) and then sin, cos or tan.

Example 3

The side opposite θ is 33 m and the hypotenuse is 105 m. Find θ.

1. Label: 33 m is O, 105 m is H.
2. Choose: O and H means sin.
3. Substitute: sin θ = 33 ÷ 105 = 0.3142…
4. Solve: θ = sin⁻¹(0.3142…) = 18.3° (1 d.p.)

Example 4

The side adjacent to θ is 12 m and the hypotenuse is 17 m. Find θ.

1. Label: 12 m is A, 17 m is H.
2. Choose: A and H means cos.
3. Substitute: cos θ = 12 ÷ 17 = 0.7058…
4. Solve: θ = cos⁻¹(0.7058…) = 45.1° (1 d.p.)

Tip: You can skip the middle step. Type SHIFT cos (12 ÷ 17) straight into your calculator to avoid rounding errors. For more time-saving buttons, see our guide to 10 calculator tricks every GCSE Maths student needs.

Common mistakes that cost marks

• Calculator in the wrong mode. Check for “D” (degrees) on the screen. A quick test is that sin 30 should equal 0.5.
• Labelling from the wrong angle. Opposite and adjacent depend on which angle you’re using, so always check before you choose a ratio.
• Rounding too early. Keep the full value on your calculator until the final answer.
• Using the normal function instead of the inverse. If you’re finding an angle, you need sin⁻¹, cos⁻¹ or tan⁻¹.
• Answers that don’t make sense. A side can never be longer than the hypotenuse, and in a right-angled triangle the other two angles must each be less than 90°.

Exact values for the non-calculator paper

You can be asked trigonometry questions on the non-calculator paper too, so both Foundation and Higher tier students need to learn these exact values:

	0°	30°	45°	60°	90°
sin	0	½	√2/2	√3/2	1
cos	1	√3/2	√2/2	½	0
tan	0	√3/3	1	√3	undefined

Remember: The sin row read backwards gives you the cos row. Learn one and you get the other for free. Our non-calculator paper tactics cover more ways to pick up marks when you can’t use a calculator.

If you’d like to go back over the basics before trying harder questions, our Trigonometry for Beginners guide spends more time on SOH CAH TOA and labelling triangles.

The four-step routine

Every right-angled trig question comes down to this:

1. Label the sides H, O and A from the given angle.
2. Choose sin, cos or tan using SOH CAH TOA.
3. Substitute your values into the formula.
4. Solve by multiplying, dividing or using the inverse function, then sense-check your answer.

Practise trigonometry with our GCSE apps

Trigonometry becomes automatic with repetition. The more triangles you label, the quicker you’ll spot which ratio to use.

• GCSE Geometry is built for this topic. It gives you visual practice on right-angled trigonometry alongside 2D and 3D shapes, angles, transformations and circle theorems, so you can drill SOH CAH TOA until labelling sides takes seconds. Try the GCSE Geometry quiz for a quick check of where you are.
• GCSE Maths is the complete revision bundle, covering Number, Algebra, Geometry and Statistics in one app. It’s ideal once you’re ready for mixed practice, where trigonometry turns up alongside Pythagoras, algebra and area in the same question.

Across the four subject apps there are 2900+ questions covering the full Foundation and Higher tier specification. Start with 10 trig questions a day and you’ll soon find the routine becomes second nature.