IGCSE Maths / Introducing trigonometry

The tangent ratio revision notes

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

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The tangent ratio links the two shorter sides of a right-angled triangle: the opposite side and the adjacent side. It helps you find a missing short side, and unlike sine and cosine, its value can be greater than 1.

What tangent measures

For an angle in a right-angled triangle, tangent is the opposite side divided by the adjacent side. The adjacent side is the short side next to the angle. The hypotenuse is not used, so tangent is useful when a question involves the two shorter sides. tan x = O/A, meaning opposite ÷ adjacent. In SOH CAH TOA, tangent is the TOA part: opposite over adjacent. If the hypotenuse is involved, use sine or cosine instead.

How to find the opposite side

When you know the adjacent side and the angle, rearrange the tangent ratio to find the opposite side. Multiply the adjacent side by the tangent of the angle.

Worked example: opposite a 28° angle

The known side is 9 cm next to the 28° angle. It is one of the two shorter sides, so use tangent, not sine or cosine.

Why tangent can be greater than 1

Sine and cosine divide by the hypotenuse, which is the longest side, so they cannot be greater than 1. Tangent divides one short side by the other, so there is no such limit. At 45°, the opposite and adjacent sides are equal. Above 45°, the opposite side is longer, making tangent greater than 1. tan 45° = 1. A tangent above 1 is not automatically a mistake.

Quick check: opposite a 45° angle

A side of 6 cm is next to a 45° angle. Since tan 45° = 1, the opposite side has the same length as the adjacent side.

Choose the ratio that matches the sides

Check what the known side represents before choosing a ratio. In the 28° example, 9 cm is next to the angle, not opposite the right angle. It is not the hypotenuse, so sine is not suitable. With the two shorter sides involved, tangent is the matching ratio. Sine and cosine both have the hypotenuse in the denominator. Use tangent when the opposite and adjacent sides are the ones involved.