IGCSE Maths / Introducing trigonometry

The cosine ratio

Edexcel International GCSE Maths (4MA1), chapter 16: Introducing trigonometry, specification point 4.8b-f. Free lesson.

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The cosine ratio links the side next to an angle in a right-angled triangle to the hypotenuse. Use it to find an unknown side when you know the angle and the hypotenuse.

What cosine measures

For an angle in a right-angled triangle, cosine is the adjacent side divided by the hypotenuse. The adjacent side is the shorter side touching the angle. The hypotenuse touches the angle too, but it is never the adjacent side. The rule is cos x = A/H, where A is the adjacent side and H is the hypotenuse. This is the CAH part of SOH CAH TOA.

How to identify the adjacent side

Every angle in a right-angled triangle touches two sides. One is the hypotenuse, so the other is the adjacent side. Which side is adjacent depends on the angle you are using. If you choose a different angle, the opposite and adjacent sides swap over. Mark the angle you are working from first. The side touching that angle, other than the hypotenuse, is the adjacent side.

How to find the adjacent side

When the hypotenuse and angle are known, multiply the hypotenuse by the cosine of the angle. This comes from rearranging cos x = A/H to make A the subject. Label the sides before choosing a ratio, so you do not confuse cosine with sine. Cosine uses the adjacent and hypotenuse. Sine uses the opposite side and hypotenuse.

When sine and cosine are equal

Sine and cosine of the same angle are not always different. At 45°, the opposite and adjacent sides are equal, so dividing each by the same hypotenuse gives equal ratios. Away from 45°, the ratios differ. sin 45° = cos 45°.

Check the side before choosing

A common error is using sine just because an angle and the hypotenuse are given. Check which side the question asks for: if it is the side touching the angle, use cosine. The adjacent side must also be shorter than the hypotenuse. For a 60° angle, cos 60° = 1/2, so the adjacent side is half the hypotenuse.

More lessons in Introducing trigonometry

• The sine ratio
• The tangent ratio
• Labelling sides and choosing the trig ratio
• Finding a missing side with trigonometry
• Missing side when the unknown is in the denominator
• Finding a missing angle with inverse trig
• Combining Pythagoras and trigonometry
• Two-step right-angled triangle problems
• Right-angled trigonometry with bearings
• Angles of elevation and depression