12.7 Finding Lengths and Angles Using Trigonometry

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: choosing sin, cos or tan, exact values, and problems mixing Pythagoras and trigonometry. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Choose the correct trigonometric ratio for any right-angled triangle problem
• Know the exact values of sin, cos and tan for 0°, 30°, 45°, 60° and 90°
• Decide whether to use Pythagoras’ theorem or trigonometry
• Solve problems that need two steps or two triangles

Key words

• SOH CAH TOA: sin = O ÷ H, cos = A ÷ H, tan = O ÷ A.
• Exact value: A value written without rounding, such as sin 30° = 1/2 or cos 30° = √3/2.
• Inverse key: sin⁻¹, cos⁻¹ or tan⁻¹: finds an angle from a ratio.
• Common side: A side shared by two triangles in a diagram, often found first.

Choosing the ratio

1. Label H, O and A from the angle. 2. Tick the side you know and the side you want (or the two you know, if you want the angle). 3. Pick the ratio that uses those two letters.

Exact values

You need to know these values without a calculator. They come from an equilateral triangle cut in half (30° and 60°) and a right-angled isosceles triangle (45°).

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

Worked example: without a calculator

A right-angled triangle has hypotenuse 14 cm and an angle of 30°. Find the side opposite the 30° angle.
O = 14 × sin 30° = 14 × 1/2 = 7

Answer: 7 cm

Key fact: sin 30° = cos 60° = 1/2 and tan 45° = 1 are the ones used most often.

Pythagoras or trigonometry?

If the question involves only sides (two known, one wanted), use Pythagoras. If an angle is given or wanted, use trigonometry.

Worked example: using both

The two shorter sides of a right-angled triangle are 7 cm and 24 cm. Find (a) the hypotenuse (b) the smallest angle.
(a) Sides only: √(7² + 24²) = √625 = 25 cm.
(b) The smallest angle is opposite the shortest side: tan θ = 7 ÷ 24, θ = tan⁻¹(7 ÷ 24) = 16.26…° = 16.3°

Answer: (a) 25 cm (b) 16.3°

Two-step problems

When there are two triangles, find the common side first using the triangle where you have enough information. Keep the full calculator value for the next step.

Worked example: two triangles

In the diagram, AB = 15 cm, angle BAD = 35°, BD is perpendicular to AC and DC = 6 cm. Find angle θ, to 1 d.p.
Triangle ABD: BD = 15 × sin 35° = 8.6036… cm (O from H).
Triangle BDC: tan θ = 8.6036… ÷ 6, so θ = tan⁻¹(1.4339…) = 55.10…° = 55.1°

Answer: 55.1°

Common mistakes

• Choosing the ratio before labelling the sides from the correct angle.
• Mixing up exact values: sin 30° = 1/2 but cos 30° = √3/2.
• Using trigonometry in a triangle without a right angle. Split it into right-angled triangles first.
• Rounding the common side in a two-step problem, which makes the final answer wrong.

Revision questions

Try these, then check your answers below each one.

Question 1. What is the exact value of cos 30°? (Choose from: 1/2, (√3)/(2), (√2)/(2), 1.)

Answer: (√3)/(2)

Working: cos 30° = (√3)/(2) ≈ 0.866.

Question 2. Without a calculator: a right-angled triangle has hypotenuse 16 cm and an angle of 60°. Work out the side adjacent to the 60° angle.

Answer: 8 cm

Working: 16 cos 60° = 16 × 1/2 = 8 cm.

Question 3. The two shorter sides of a right-angled triangle are 5 cm and 9 cm. Work out the smallest angle, correct to 1 decimal place.

Answer: 29.1 °

Working: tan⁻¹(5 ÷ 9) = 29.05… ≈ 29.1°.

Related lessons

• Previous lesson: 12.6 Trigonometry: the Tangent Ratio
• Next lesson: 13.1 Calculating Probability
• In this chapter: 12.1 Pythagoras' Theorem 1
• In this chapter: 12.2 Pythagoras' Theorem 2
• In this chapter: 12.3 Trigonometry: the Sine Ratio 1

Continue in the full course

This lesson is section 12.7 (EGCMF1207) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.