Opposite, adjacent and hypotenuse
Most trigonometry marks are lost before any calculation happens. They are lost at the labelling step.
A student writes SOH CAH TOA at the top of the page, picks sine, and then divides the wrong two numbers. The method was right. The labels were wrong.
Here is the order that fixes it.
Step 1: find the right angle
Look for the small square in one corner. That is the right angle, 90 degrees.
If there is no square, check the question. A triangle is only right-angled if it says so, or if it is drawn with one corner clearly square and marked.
Step 2: the hypotenuse never moves
The hypotenuse is the side opposite the right angle.
It is always the longest side. It is always the same side, no matter which angle the question asks about. Label it first, and label it once.
That is the whole reason to start here. One of the three labels is fixed. Get it down and you only have two left to sort out.
Step 3: stand at the angle in the question
This is the step people skip.
A right-angled triangle has two other angles. The question will name one of them, often with a symbol like theta, or a letter such as A or x. Everything that follows depends on which one.
Put your finger on that angle. Now:
- The opposite side is the one you are not touching. It sits across the triangle from your finger.
- The adjacent side is the one you are touching that is not the hypotenuse. Adjacent means next to.
Two sides touch your angle: the hypotenuse and the adjacent. One does not: the opposite. That is the entire rule.
Step 4: check it by moving
Here is the test that proves you understand it.
Move your finger to the other acute angle. The hypotenuse stays exactly where it was. The opposite and the adjacent swap over.
If that surprises you, read step 3 again. If it does not, you have it.
This is why a student can do the same triangle twice and get two different answers. They used one angle for the labels and the other angle in the calculation.
Putting it together
Only now does SOH CAH TOA mean anything:
- Sine = Opposite over Hypotenuse
- Cosine = Adjacent over Hypotenuse
- Tangent = Opposite over Adjacent
Read the question, see which two sides it gives you or asks for, and pick the ratio that uses exactly those two.
Given the opposite and the hypotenuse? That is sine. Given the adjacent and the hypotenuse? Cosine. Given the opposite and the adjacent, and no hypotenuse anywhere? Tangent.
A worked example
A ladder leans against a wall. It is 5 metres long and makes a 60 degree angle with the ground. How far up the wall does it reach?
- The right angle is where the wall meets the ground.
- The hypotenuse is the ladder, 5 metres, because it is opposite the right angle.
- Stand at the 60 degree angle, on the ground. The wall is the side you are not touching, so the wall is the opposite. The ground is the adjacent.
- We want the opposite, and we have the hypotenuse. Opposite over hypotenuse is sine.
So sine of 60 equals the height divided by 5. The height is 5 times sine of 60, which is about 4.33 metres.
Notice that step 3 did all the work. The calculation was one line.
Where students go wrong
They label before reading the angle. The labels are meaningless until you know which angle you are standing at.
They assume the longest drawn side is the hypotenuse. Diagrams are not to scale. Find the right angle and go opposite it.
They forget to relabel in multi-step questions. Part (b) often uses the other angle. New angle, new labels.
They mix up adjacent and hypotenuse. Both touch the angle. The hypotenuse is the one opposite the right angle; the adjacent is the other one.
Practise it on its own
Labelling is a separate skill from calculating, so practise it on its own.
Draw ten right-angled triangles. Mark a different angle in each. Label all three sides. Do not calculate anything. Check them, then do ten more with the other angle marked.
Ten minutes of that beats an hour of full questions, because it fixes the step where the marks were actually going.
FutureSchool teaches trigonometry this way in its Maths programme: the labelling step gets its own video lesson, its own worksheet and its own test, before any ratio appears. Trigonometry starts in the middle secondary years and runs through to GCSE.
Not sure which year your child is actually working at? The free assessment finds their level in about twenty minutes and tells you which topics are secure and which are not.
Questions parents ask
Which side is the hypotenuse?
The longest side, always opposite the right angle. It never changes, whichever angle you are working with.
How do I tell the opposite from the adjacent side?
Both depend on which angle you are using. Stand at that angle. The side you are not touching is the opposite. The side you are touching, that is not the hypotenuse, is the adjacent.
Why does my answer change when I swap the angles?
Because opposite and adjacent swap over. The hypotenuse stays put, but the other two labels trade places when you move to the other acute angle. Relabel before you calculate.
What does SOH CAH TOA mean?
Sine equals Opposite over Hypotenuse. Cosine equals Adjacent over Hypotenuse. Tangent equals Opposite over Adjacent. It only works once the three sides are labelled correctly.