Right-Angled Trigonometry (SOHCAHTOA)
Every right-angled triangle hides a relationship between its angles and its sides. Trigonometry gives you three simple ratios to unlock it: sine, cosine, and tangent. Learn how to label the sides, choose the right ratio using SOHCAHTOA, and find missing sides or angles step by step. This is one of the most important topics in IGCSE maths, and once it clicks, it opens the door to everything from 3D problems to calculus.
Labelling the Sides of a Right-Angled Triangle
Before using any trigonometric ratio, you need to identify three sides relative to the angle you are working with (not the right angle):
The hypotenuse is always the longest side, sitting opposite the right angle. It never changes. The opposite and adjacent sides depend on which angle you are looking at: the opposite is directly across from your angle, and the adjacent is next to it.
The Three Trigonometric Ratios
Each ratio links an angle in a right-angled triangle to a pair of sides. Together they are remembered using the mnemonic SOHCAHTOA:
Use when you have, or need, the opposite and the hypotenuse.
Use when you have, or need, the adjacent and the hypotenuse.
Use when you have, or need, the opposite and the adjacent.
How to Choose the Right Ratio
- Label the sides as Opposite, Adjacent, and Hypotenuse relative to the angle in the question.
- Identify which two sides are involved (one you know, one you need, or two you know when finding an angle).
- Pick the ratio that connects those two sides: SOH, CAH, or TOA.
- Substitute and solve. If finding a side, rearrange the formula. If finding an angle, use the inverse function (\(\sin^{-1}\), \(\cos^{-1}\), or \(\tan^{-1}\)).
Finding a Side
💡 Example 1
Find the length of side \(x\), correct to 3 significant figures.
Relative to 38°:
\(x\) = Opposite, 14 = Hypotenuse
Use SOH:
\[\sin 38° = \frac{x}{14}\] \[x = 14 \times \sin 38°\] \[x = 8.62 \text{ cm (3 s.f.)}\]Why SOH?
We have the hypotenuse (14) and need the opposite (\(x\)). The ratio linking Opposite and Hypotenuse is Sine.
💡 Example 2
Find the length of side \(y\), correct to 3 significant figures.
Relative to 52°:
\(y\) = Adjacent, 20 = Hypotenuse
Use CAH:
\[\cos 52° = \frac{y}{20}\] \[y = 20 \times \cos 52°\] \[y = 12.3 \text{ cm (3 s.f.)}\]Why CAH?
We have the hypotenuse (20) and need the adjacent (\(y\)). The ratio linking Adjacent and Hypotenuse is Cosine.
💡 Example 3
Find the length of side \(p\), correct to 3 significant figures.
Relative to 65°:
\(p\) = Opposite, 9 = Adjacent
Use TOA:
\[\tan 65° = \frac{p}{9}\] \[p = 9 \times \tan 65°\] \[p = 19.3 \text{ cm (3 s.f.)}\]Why TOA?
We have the adjacent (9) and need the opposite (\(p\)). The ratio linking Opposite and Adjacent is Tangent.
💡 Example 4
Find the length of side \(h\), correct to 3 significant figures.
Relative to 28°:
7 = Opposite, \(h\) = Hypotenuse
Use SOH:
\[\sin 28° = \frac{7}{h}\] \[h \times \sin 28° = 7\] \[h = \frac{7}{\sin 28°}\] \[h = 14.9 \text{ cm (3 s.f.)}\]Unknown on the bottom
When the unknown is the denominator of the fraction, multiply both sides by \(h\) first, then divide by the trig value. This is a common pattern students need to practise.
Finding an Angle
When you know two sides and need to find an angle, use the inverse trig function. Your calculator labels these as \(\sin^{-1}\), \(\cos^{-1}\), and \(\tan^{-1}\) (sometimes written as arcsin, arccos, arctan).
💡 Example 5
Find angle \(\theta\), correct to 1 decimal place.
Relative to \(\theta\):
6 = Opposite, 11 = Adjacent
Use TOA:
\[\tan\theta = \frac{6}{11}\] \[\theta = \tan^{-1}\!\left(\frac{6}{11}\right)\] \[\theta = 28.6°\text{ (1 d.p.)}\]Using your calculator
Press SHIFT then tan to access \(\tan^{-1}\). Type the fraction, close the bracket, and press =.
💡 Example 6
Find angle \(\alpha\), correct to 1 decimal place.
Relative to \(\alpha\):
5 = Opposite, 13 = Hypotenuse
Use SOH:
\[\sin\alpha = \frac{5}{13}\] \[\alpha = \sin^{-1}\!\left(\frac{5}{13}\right)\] \[\alpha = 22.6°\text{ (1 d.p.)}\]Why SOH?
We have the opposite (5) and the hypotenuse (13). The ratio connecting those two is Sine.
Angles of Elevation and Depression
In real-world problems, you will often see two special angles:
The angle of elevation is measured upwards from the horizontal to the line of sight. The angle of depression is measured downwards from the horizontal to the line of sight. Both are always measured from the horizontal, never from the vertical.
💡 Example 7: Angle of Elevation
A boat is 85 m from the base of a cliff. The angle of elevation of the top of the cliff from the boat is 42°. Find the height of the cliff, correct to 3 significant figures.
The height is opposite the 42° angle. The 85 m is the adjacent side.
Use TOA:
\[\tan 42° = \frac{h}{85}\] \[h = 85 \times \tan 42°\] \[h = 76.5 \text{ m (3 s.f.)}\]Sketch it first
Always draw a quick right-angled triangle from the information given. The cliff is the vertical (opposite), the ground distance is the horizontal (adjacent), and the angle of elevation sits at the observer's position.
🔑 Key Points
Trigonometry only works in right-angled triangles. Always check for the right angle first.
The hypotenuse is always opposite the right angle. It is always the longest side.
Label the sides relative to the angle you are working with, not the right angle.
When the unknown is on the bottom of the fraction, rearrange by multiplying both sides by the unknown, then dividing by the trig value.
Make sure your calculator is in degree mode, not radians.
⚠️ Common Mistakes
Mixing up opposite and adjacent. Always ask: "opposite to which angle?" and "adjacent to which angle?"
Using the wrong ratio. A quick check: if your answer for a side is larger than the hypotenuse, something has gone wrong.
Forgetting to use the inverse function when finding an angle. \(\sin^{-1}\) undoes \(\sin\); it is not the same as \(\frac{1}{\sin}\).
Rounding too early. Keep all decimal places on your calculator until the final answer, then round.