IGCSE Mathematics: Trigonometry Practice Questions
Right-angled trigonometry links an angle to two sides of a triangle through sine, cosine and tangent. Label the sides opposite, adjacent and hypotenuse relative to the angle you are using, then choose the ratio that contains the two sides you know or want.
Topic 6 of Cambridge IGCSE Mathematics 0580 is one of the most reliable sources of marks on Papers 2 and 4, provided you label the triangle before you calculate. This page covers right-angled trigonometry, which applies only to triangles containing a 90 degree angle. Non-right-angled triangles need the sine and cosine rules, which are a separate topic.
What you need to know for Trigonometry
- Labelling the triangleThe hypotenuse is always opposite the right angle. The opposite side is across from the angle you are using. The adjacent side is next to that angle and is not the hypotenuse.
- SOH CAH TOAsin = opposite / hypotenuse, cos = adjacent / hypotenuse, tan = opposite / adjacent.
- Finding an angleUse the inverse functions on your calculator: sin-1, cos-1 and tan-1. Make sure the calculator is in degree mode.
- Angle of elevation and depressionBoth are measured from the horizontal. Elevation is upwards from the horizontal, depression is downwards from the horizontal.
- BearingsMeasured clockwise from north, always written with three figures, for example 072 degrees.
- AccuracyKeep full calculator accuracy through the working and round only at the end, normally to three significant figures unless the question says otherwise.
IGCSE Mathematics Trigonometry questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
A right-angled triangle has a hypotenuse of 12 cm. One of the acute angles is 35 degrees. Calculate the length of the side opposite that angle. Give your answer to 3 significant figures.
Show the worked answer
- Label the sides. The hypotenuse is 12 cm and the unknown side is opposite the 35 degree angle.
- Opposite and hypotenuse means use sine: sin 35 = opposite / 12.
- Rearrange: opposite = 12 x sin 35.
- opposite = 12 x 0.5736 = 6.883, which is 6.88 cm to 3 significant figures.
In a right-angled triangle the side opposite angle A is 7.0 cm and the side adjacent to angle A is 9.0 cm. Calculate angle A to 1 decimal place.
Show the worked answer
- Opposite and adjacent are known, so use tangent: tan A = opposite / adjacent.
- tan A = 7.0 / 9.0 = 0.7778.
- Take the inverse: A = tan-1(0.7778).
- A = 37.87 degrees, which is 37.9 degrees to 1 decimal place.
A ladder of length 5.0 m rests against a vertical wall. The foot of the ladder is 1.4 m from the base of the wall. Calculate the angle between the ladder and the ground, and the height the ladder reaches up the wall. Give both answers to 2 significant figures.
Show the worked answer
- The ladder is the hypotenuse, 5.0 m. The distance along the ground, 1.4 m, is adjacent to the angle at the ground.
- Adjacent and hypotenuse means cosine: cos A = 1.4 / 5.0 = 0.28, so A = cos-1(0.28) = 73.74 degrees, which is 74 degrees to 2 significant figures.
- For the height, use Pythagoras: height2 = 5.02 - 1.42 = 25 - 1.96 = 23.04.
- height = square root of 23.04 = 4.80 m, which is 4.8 m to 2 significant figures.
A surveyor stands 40 m from the base of a tower on level ground. Her eyes are 1.6 m above the ground and the angle of elevation of the top of the tower from her eyes is 28 degrees. Calculate the height of the tower to 3 significant figures.
Show the worked answer
- The angle of elevation is measured from the horizontal line at eye level, so the triangle has an adjacent side of 40 m.
- Opposite and adjacent means tangent: tan 28 = h / 40, where h is the height above eye level.
- h = 40 x tan 28 = 40 x 0.5317 = 21.27 m.
- Add the height of her eyes: total height = 21.27 + 1.6 = 22.87, which is 22.9 m to 3 significant figures.
Common mistakes in this topic
- Labelling the sides before choosing the angle. The opposite and adjacent sides swap depending on which acute angle you use.
- Leaving the calculator in radian or gradian mode.
- Using trigonometry on a triangle with no right angle. That needs the sine or cosine rule instead.
- Rounding partway through and carrying the rounded value into the next step.
- Forgetting the units, or giving an angle without the degree symbol.
Exam tips
- Draw the triangle and mark the right angle first, then label H, O and A relative to the angle in the question. Do this even when a diagram is given.
- Write the ratio equation before substituting numbers. That line is usually worth a method mark on its own.
- Sanity check your answer. The hypotenuse must be the longest side, and the two acute angles must add to 90 degrees.
- If you are given two sides and want an angle, you need an inverse function. If you are given an angle and one side, you do not.
- For elevation and depression problems, sketch the horizontal line first. Almost every error in these questions comes from measuring the angle from the wrong line.
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Trigonometry FAQs
What does SOHCAHTOA mean?
It is a memory aid for the three trigonometric ratios in a right-angled triangle. SOH means sine equals opposite over hypotenuse. CAH means cosine equals adjacent over hypotenuse. TOA means tangent equals opposite over adjacent. Label the triangle first, then pick the ratio containing the two sides you know or want.
How do I find an angle rather than a side?
Form the ratio using the two sides you know, then apply the inverse function on your calculator. For example, if the opposite side is 7 and the adjacent is 9, then tan A equals 7 divided by 9, and A equals the inverse tangent of that value, which is 37.9 degrees to one decimal place.
Can I use SOHCAHTOA on any triangle?
No. The three ratios only apply to right-angled triangles. For a triangle with no right angle you need the sine rule or the cosine rule, or you can split the triangle into two right-angled triangles by dropping a perpendicular.
What is the difference between angle of elevation and angle of depression?
Both are measured from the horizontal. The angle of elevation is the angle you look up through from the horizontal to see an object above you. The angle of depression is the angle you look down through from the horizontal to see an object below you. Always draw the horizontal line before marking the angle.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.