IGCSE Maths / Introducing trigonometry

Labelling sides and choosing the trig ratio

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

Read the revision notes · IGCSE Maths past papers

Label the sides of a right-angled triangle from the angle you are using, then choose sine, cosine or tangent to match the two sides in the question. This helps you set up the correct ratio before calculating.

How to find the hypotenuse

The hypotenuse is the longest side of a right-angled triangle. Find it by looking for the side facing the right angle. It is the only side that does not touch the right angle, so its identity does not change if the triangle is drawn in a different position. Do not identify it just by looking for the bottom or sloping side. Its short name is H.

How to label opposite and adjacent

The names opposite and adjacent depend on which acute angle you are working from. Mark that angle first, then label the other sides in relation to it. Opposite (O) faces the angle you are using. Adjacent (A) joins your angle to the right angle. It is the side left over after you identify the hypotenuse and opposite. If you switch to the other acute angle, opposite and adjacent swap places. The hypotenuse stays the same.

How SOH CAH TOA matches the sides

SOH CAH TOA helps you remember which pair of sides goes with each ratio. In each fraction, use the side names from the angle you are working from. The angle is often written as θ, pronounced theta. SOH: sin θ = opposite/hypotenuse CAH: cos θ = adjacent/hypotenuse TOA: tan θ = opposite/adjacent These ratios use acute angles, which are angles below 90°.

How to choose a ratio and set it up

Label the sides before putting any numbers into a ratio. Then name the two sides you have or need, and choose the ratio that contains that pair.

Worked example: finding an opposite side

Triangle ABC is right-angled at C. Angle A is 35° and AC = 8 cm. Find BC. From angle A, AB is the hypotenuse, BC is opposite and AC is adjacent. The pair is opposite and adjacent, so use tangent.

A quick choice and the key reminders

If you know the hypotenuse and want the side opposite a 40° angle, the pair is opposite and hypotenuse. That means you need sine. sin 40° = opposite/hypotenuse SOH CAH TOA is not printed on the Edexcel formula sheet, so learn it and write the ratio out before substituting numbers.

More lessons in Introducing trigonometry

• The sine ratio
• The cosine ratio
• The tangent 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