Study Tips
How to Understand Trigonometry (Not Just Memorise It)
By Sana Iqbal · · 4 min read
Quick answer
Sine, cosine and tangent are ratios of a right-angled triangle's sides. SOHCAHTOA tells you which ratio uses which sides. To choose the right one, label the sides relative to the angle you know or want — opposite, adjacent, hypotenuse — then pick the ratio that uses the two sides you have.
The ratios, and what they mean
In a right-angled triangle, sine, cosine and tangent are each a fixed ratio between two sides for a given angle. The same angle always gives the same ratios, no matter how big the triangle — which is what makes them useful.
That constancy is the whole idea: measure an angle and one side, and the ratios let you find the others. Trigonometry is a machine for converting between angles and lengths.
SOHCAHTOA, used properly
SOHCAHTOA reminds you which sides each ratio uses: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.
The trick students miss is that opposite and adjacent are defined relative to the angle you are working with, not fixed to the triangle. Relabel the sides for each new angle before choosing a ratio.
Choosing the right ratio
Label the two sides that matter — the one you know and the one you want — as opposite, adjacent or hypotenuse relative to your angle. Then pick the ratio that uses exactly those two.
This removes the guessing. If you have opposite and hypotenuse, it is sine; adjacent and hypotenuse, cosine; opposite and adjacent, tangent. No other choice fits.
Finding an angle versus a side
To find a side, use the ratio directly. To find an angle, use the inverse function on your calculator — the sine, cosine or tangent 'minus one' button — which turns a ratio back into the angle.
Students often forget the inverse and get a nonsensical answer. If you are solving for an angle, you almost always need the inverse function.
Beyond right angles
For triangles without a right angle, the sine rule and cosine rule take over. Use the sine rule when you have a matching angle-and-side pair, and the cosine rule when you have two sides and the angle between them, or all three sides.
Knowing which rule applies is most of the difficulty. Write down what you have — which sides, which angles — and the choice usually becomes clear.
The unit circle later
At A-Level, trigonometry expands beyond triangles to the unit circle, which is how sine and cosine become smooth repeating waves. That is where the graphs, identities and radians come from.
It looks like a new subject and it is the same ratios extended. Keeping the triangle meaning in mind stops the identities feeling arbitrary.
Frequently asked questions
How do I know whether to use sine, cosine or tangent?+
Label the sides you know and want relative to your angle — opposite, adjacent, hypotenuse — then pick the ratio that uses those two sides. There is only one that fits.
What's the difference between sin and sin inverse?+
Sine turns an angle into a ratio; the inverse (sin⁻¹) turns a ratio back into an angle. Use the normal function to find a side and the inverse to find an angle.
When do I use the sine rule or cosine rule?+
For triangles without a right angle. Use the sine rule when you have a matching angle and opposite side, and the cosine rule for two sides and the included angle, or all three sides.
Why does the same angle always give the same ratio?+
Because triangles with the same angles are similar — scaled versions of each other — so the ratios of their sides are identical regardless of size. That constancy is what makes trigonometry work.
What are radians and why use them?+
Radians measure angles by arc length rather than degrees, and they make calculus with trigonometric functions far simpler. You meet them at A-Level alongside the unit circle.
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