IGCSE Trigonometry: How to Choose Sine, Cosine or Tangent

Audience and search intent: Cambridge IGCSE Mathematics 0580 students (Core and Extended), parents and tutors searching for a first practical explanation of right-angled triangle trigonometry.

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The three ratios solve different questions

A right-angled triangle contains one 90° angle. If we know an acute angle and one side, trigonometry can help us find another side. If we know two sides, it can help us find an acute angle. The important skill is not reciting a rhyme; it is choosing the relationship that matches the information given.

For Cambridge IGCSE Mathematics 0580 examinations in 2025–2027, the sine, cosine and tangent ratios for acute angles in right-angled triangles appear in both Core (C6.2) and Extended (E6.2). More advanced trigonometric content, such as exact values and trigonometric graphs, is Extended-only. That distinction matters when choosing revision material.

Step 1: Label the triangle relative to the angle

Imagine a right-angled triangle and select one of its two acute angles, called $\theta$ (theta). The hypotenuse is the side opposite the right angle and is always the longest side. The opposite side is across from $\theta$. The adjacent side touches $\theta$ but is not the hypotenuse.

The labels “opposite” and “adjacent” change if you choose the other acute angle. The hypotenuse does not. Before substituting numbers into a formula, mark the angle and identify the three sides on the drawing.

Step 2: Choose a ratio by the sides you need

The three definitions are:

• Sine: $\sin\theta = \text{opposite}/\text{hypotenuse}$.

• Cosine: $\cos\theta = \text{adjacent}/\text{hypotenuse}$.

• Tangent: $\tan\theta = \text{opposite}/\text{adjacent}$.

SOH CAH TOA is a memory aid for these three ratios. It does not tell you which to use until you identify the two relevant sides. If the problem gives the adjacent side and asks for the opposite, tangent is the direct choice; if it gives the hypotenuse and asks for the opposite, use sine.

Worked example 1: Find a height using tangent

A triangular support has a horizontal base of 12 m and an angle of 35° between the base and its sloping side. The upright height is perpendicular to the ground. Find the height, to two decimal places.

Relative to 35°, the height is opposite and the 12 m base is adjacent. Therefore:

$$\tan 35^\circ=\frac{h}{12}$$

Multiply by 12:

$$h=12\tan 35^\circ\approx 8.40\text{ m}.$$

Reasonableness check: The angle is less than 45°, so the opposite side should be shorter than the 12 m adjacent side. A height of 8.40 m fits the picture. A result of 17 m would be a signal to recheck the ratio or calculator.

Worked example 2: Find a side using cosine

Using the same triangle, find the length $s$ of the sloping side. The base of 12 m is adjacent to 35°, and $s$ is the hypotenuse.

$$\cos35^\circ=\frac{12}{s}$$

Rearrange to obtain:

$$s=\frac{12}{\cos35^\circ}\approx14.65\text{ m}.$$

A hypotenuse must be longer than either of the other sides, so 14.65 m is plausible. Notice that dividing by a number smaller than 1 produces a value larger than 12.

Worked example 3: Find an angle, not a length

A triangular ramp rises 7.5 cm for every 10 cm of horizontal distance. What is its angle above the horizontal?

The rise is opposite and the run is adjacent:

$$\tan\theta=\frac{7.5}{10}=0.75.$$

To recover the angle, use the inverse tangent function on a calculator:

$$\theta=\tan^{-1}(0.75)\approx36.9^\circ.$$

Here $\tan^{-1}$ means the inverse trigonometric function, not $1/\tan$. Check that the calculator is in degree mode, not radian mode. Cambridge's 2025–2027 syllabus specifies degree answers and, when decimal angles are required, one decimal place.

Three mistakes that cost marks

Mistake 1: Choosing the side names before choosing the angle. Opposite and adjacent are relative labels. Mark the reference angle first.

Mistake 2: Using Pythagoras when the only known measurements are an angle and one side. Pythagoras relates three side lengths; a trig ratio is the natural starting point when an acute angle and a side are given.

Mistake 3: Forgetting the context and units. An angle answer needs degrees; a length answer needs units. When finding a side, ask whether it should be shorter or longer than the hypotenuse.

Trigonometry here assumes a right-angled triangle. Do not automatically use SOHCAHTOA on an arbitrary triangle. Extended students later encounter other methods for non-right-angled triangles, including the sine and cosine rules.

Four original IGCSE-style practice questions

1. A right-angled triangle has hypotenuse 15 cm and an acute angle of 28°. Find the side opposite that angle, to two decimal places.

2. The opposite side is 4 m and the adjacent side is 9 m. Find the acute angle, to one decimal place.

3. A side opposite a 40° angle is 11 m. Find the hypotenuse, to two decimal places.

4. A straight ramp rises 0.9 m over a horizontal run of 7.2 m. Find its angle above the horizontal, to one decimal place.

Answers and reasoning

1. $15\sin28^\circ\approx\mathbf{7.04\text{ cm}}$. Opposite and hypotenuse imply sine.

2. $\tan^{-1}(4/9)\approx\mathbf{24.0^\circ}$. Opposite and adjacent imply tangent.

3. $11/\sin40^\circ\approx\mathbf{17.11\text{ m}}$. The hypotenuse is correctly longer than 11 m.

4. $\tan^{-1}(0.9/7.2)\approx\mathbf{7.1^\circ}$. The shallow rise produces a small angle.

How parents can help without teaching every formula

Give your child three unlabelled right-angled triangles with the reference angle in different positions. Ask them to label opposite, adjacent and hypotenuse before showing any numbers. Next, give only the two relevant side names and ask whether sine, cosine or tangent is appropriate. Finally, let them solve one length and one angle problem and check the answer against the sketch.

A student who can justify why a ratio matches the known and unknown sides is better prepared for unfamiliar IGCSE questions than one who can only repeat SOHCAHTOA.

Official fact-check sources (consulted 29 October 2026): Cambridge IGCSE Mathematics 0580 syllabus for 2025–2027, Topic 6, C6.2 and E6.2; Cambridge IGCSE Mathematics qualification page. The four questions above are original illustrative questions, not Cambridge past-paper reproductions.

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