IGCSE Maths / Introducing trigonometry

Finding a missing side with trigonometry

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

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Use the angle and the sides you know to choose sine, cosine or tangent, then rearrange the ratio to find the missing side. Check calculator mode and round your answer to 3 significant figures at the end.

Label the sides from the angle

In a right-angled triangle, label the sides from the acute angle you are using. The hypotenuse is opposite the right angle. The opposite side faces your chosen angle, and the adjacent side is next to it. The pair of sides in the question tells you which trigonometric ratio to use. sin = opposite ÷ hypotenuse cos = adjacent ÷ hypotenuse tan = opposite ÷ adjacent

Find a side using sine

In triangle ABC, the right angle is at C. From angle A, BC faces the angle, so it is the opposite side. AB is opposite the right angle, so it is the hypotenuse. Opposite and hypotenuse make a sine ratio. Since BC is on top of the fraction, multiply by 11 to make BC the subject.

Choose the correct ratio

Cosine also has the hypotenuse underneath, but it pairs it with the adjacent side, not the opposite. BC faces the 42° angle, so it is opposite. Choosing cosine instead of sine gives the wrong ratio and loses the marks for the method. For opposite and hypotenuse, use sine. For adjacent and hypotenuse, use cosine. For opposite and adjacent, use tangent.

Find a side using cosine

Here, AC is next to angle A, so it is the adjacent side. AB is still the hypotenuse. Use cosine for this pair. The answer should be shorter than the 9.4 cm hypotenuse, which is a useful check.

Let the position of the unknown guide you

Finding a missing side does not always mean multiplying. First write the ratio, then see where the unknown is. If it is on top, multiply to make it the subject. If it is underneath, divide. The rearrangement, not the topic, decides the final operation. Check that your calculator is in degree mode. For example, sin 42° should show 0.669. Keep the full calculator value, then round once at the end to 3 significant figures.

Use tangent when the sides are opposite and adjacent

In triangle PQR, the right angle is at Q. From angle P, PQ is next to the angle and QR faces it. This is the adjacent and opposite pair, so use tangent. QR is on top of the ratio, so multiply by 7.

More lessons in Introducing trigonometry

• The sine ratio
• The cosine ratio
• The tangent ratio
• Labelling sides and choosing the trig ratio
• 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