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12.5 Trigonometry: the Cosine Ratio

GCSE Maths Foundation · Edexcel · Chapter 12: Right-angled triangles

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: the cosine ratio: finding the adjacent side, the hypotenuse and an angle. 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

  • Know that cos θ = adjacent ÷ hypotenuse
  • Use cos θ to find the adjacent side or the hypotenuse
  • Use cos⁻¹ to find an angle
  • Choose between sine and cosine

Key words

  • Cosine (cos): For a marked angle in a right-angled triangle, cos θ = adjacent ÷ hypotenuse.
  • Adjacent (A): The side next to the marked angle that is not the hypotenuse.
  • Inverse cosine (cos⁻¹): Turns a value of cos θ back into the angle. Usually SHIFT then cos.
  • CAH: Memory aid: Cosine = Adjacent over Hypotenuse.

The cosine ratio

Cosine uses the two sides that touch the angle θ: the adjacent and the hypotenuse.

Key fact: cos θ = A ÷ H so A = H × cos θ, H = A ÷ cos θ, θ = cos⁻¹(A ÷ H)

Worked example: values to know

Write down cos 0°, cos 60° and cos 90°.
From the graph (or the calculator): cos 0° = 1, cos 60° = 0.5, cos 90° = 0.

Answer: 1, 0.5 and 0

Finding the adjacent side

If you know H and θ, the adjacent side is A = H × cos θ.

Finding the hypotenuse or an angle

If the hypotenuse is unknown, divide. If the angle is unknown, use cos⁻¹.

Worked example: the hypotenuse

The side adjacent to a 41° angle is 6.5 cm. Find the hypotenuse.
cos 41° = 6.5 ÷ H, so H = 6.5 ÷ cos 41° = 8.612… = 8.61 cm

Answer: 8.61 cm

Worked example: an angle

The adjacent side is 5 cm and the hypotenuse is 8 cm. Find θ.
cos θ = 5 ÷ 8 = 0.625. θ = cos⁻¹(0.625) = 51.317…° = 51.3°

Answer: 51.3°

Sine or cosine?

Label the sides from θ. Tick the two sides you know or want. If they are O and H, use sine. If they are A and H, use cosine.

  • opposite and hypotenuse; Use sine; Formula sin θ = O ÷ H
  • adjacent and hypotenuse; Use cosine; Formula cos θ = A ÷ H

Worked example: a ladder

A 6 m ladder makes 74° with the ground. How far is its foot from the wall?
The ladder is H. The ground distance touches the 74° angle, so it is A: use cosine.
A = 6 × cos 74° = 1.653… = 1.65 m

Answer: 1.65 m

Common mistakes

  • Using sine when the side you need touches the angle. Touching (and not H) means adjacent: use cos.
  • Multiplying when the hypotenuse is the unknown. Then H = A ÷ cos θ.
  • Pressing cos instead of cos⁻¹ when finding an angle.
  • Forgetting that cos θ gets smaller as θ gets bigger, so a bigger angle gives a shorter adjacent side.

Revision questions

Try these, then check your answers below each one.

Question 1. Use your calculator to work out cos 25°, correct to 4 decimal places.

Answer: 0.9063

Working: cos 25° = 0.90630… ≈ 0.9063.

Question 2. A right-angled triangle has hypotenuse 12 cm and an angle of 40°. Work out the side adjacent to the 40° angle, correct to 3 s.f.

Answer: 9.19 cm

Working: 12 cos 40° = 9.1925… ≈ 9.19 cm.

Question 3. The side adjacent to a 65° angle in a right-angled triangle is 4 cm. Work out the hypotenuse, correct to 3 s.f.

Answer: 9.46 cm

Working: 4 ÷ cos 65° = 9.4648… ≈ 9.46 cm.

Related lessons

Continue in the full course

This lesson is section 12.5 (EGCMF1205) 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.

 

 

 

 

 

 

 

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