Opposite and adjacent are not a property of a side but of the side together with the angle you are looking at. If you know two sides and want the third, you use Pythagoras' theorem. If an angle joins in, you choose sine, cosine or tangent on the two sides you have.
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In a right-angled triangle the opposite side is fixed. Most people say yes, and that costs marks as soon as you resolve forces.
In just over three minutes: the hypotenuse, and why opposite and adjacent are not a property of a side but of the side together with the angle you are looking at. Look at the other angle and those two simply swap round. Then Pythagoras' theorem for when you know two sides, and sine, cosine and tangent for when an angle joins in, where you choose which of the three to take by looking at which two sides you have. Not the other way round. Finally one sum worked out in full, with the check on whether the answer can be right and with the mistake that sits underneath it if you divide instead of multiply.
For upper secondary physics.
In this video
The full explanation, in writing
The problem
0:00In a right-angled triangle the opposite side is fixed. Is that right, yes or no? Most people say yes, and that costs marks as soon as you resolve forces.
The names in the triangle
0:14Then the triangle. There is one angle of ninety degrees, and the side opposite it is the longest: the hypotenuse. The two other sides have no fixed name, because their name depends on which angle you look at. Look at this angle, this side is the opposite and that one the adjacent.
0:37Look at the other angle, those two simply swap round. That is the mistake that is made most often: opposite and adjacent are not properties of the side, but of the combination side-and-angle.
Pythagoras, sine, cosine and tangent
0:56Know two sides, the third follows from Pythagoras' theorem: the two shorter sides squared and added is the hypotenuse squared. If you want an angle too, you have three ratios. The sine of an angle is the opposite divided by the hypotenuse. The cosine is the adjacent divided by the hypotenuse.
1:21And the tangent is the opposite divided by the adjacent. You choose which of the three to take by looking at which two sides you have. Not the other way round.
One sum, step by step
1:35One sum. The hypotenuse is fifteen centimetres, and this angle is thirty-two degrees. How long is the opposite side? You have the hypotenuse, you want the opposite. That pair belongs with the sine. The sine of thirty-two degrees is the opposite divided by fifteen.
1:55So the opposite is fifteen times the sine of thirty-two, and that is seven point nine centimetres. Just check whether that can be: the opposite must be shorter than the hypotenuse, and seven point nine is indeed smaller than fifteen. If you get twenty-eight, you have divided instead of multiplied.
2:22And the third side you can get two ways: fifteen times the cosine of thirty-two, or Pythagoras on the two you now have. Both twelve point seven. Opposite and adjacent depend on which angle you look at, not on the side itself. Two sides known and you want the third: Pythagoras.
2:45If you want an angle too, then it is sine, cosine or tangent, and you choose on the two sides you already have. One question to take with you: you have the adjacent and the opposite, and you want the angle. Which of the three do you take? The tangent. That is the only one of the three that does not contain the hypotenuse.
3:10Look at which two sides you have. Then the ratio chooses itself.
Frequently asked questions
When do you use sine, cosine or tangent?
You choose on the two sides involved. Opposite and hypotenuse is the sine, adjacent and hypotenuse is the cosine, opposite and adjacent is the tangent. So you look at which two sides you have, not at which formula you remember.
Which ratio do you take if you have the adjacent and the opposite and want the angle?
The tangent. It is the only one of the three that does not contain the hypotenuse, so it is the only one you can use when you have those two sides.
Do opposite and adjacent swap round if you look at the other angle?
Yes. They are not properties of the side but of the side together with the angle you are looking at. Look at the other angle and the two simply swap round. That is the mistake made most often.
The hypotenuse is 15 cm and the angle is 32 degrees. How long is the opposite side?
Seven point nine centimetres. You have the hypotenuse and you want the opposite, so that is the sine: the opposite is fifteen times the sine of thirty-two. Check it: the opposite must be shorter than the hypotenuse, and 7.9 is indeed smaller than 15.
How do you find the third side if you know the hypotenuse and one angle?
Two ways, and both should give the same answer. Fifteen times the cosine of thirty-two, or Pythagoras on the two sides you now have. Both give twelve point seven.
Physics with Thomas makes physics explainers for upper secondary school. No words on screen, so what you see works in any language.
In this series
Quantities and unitswhy 20,000 is not more than 10,0007:30
Ratios and percentagesone table for both3:15
Scientific notationpowers of ten and order of magnitude5:22
Calculating with powers of tenthe four rules3:18
Pythagoras and trigonometrywhich side with which angle3:19
Deriving units from a formulaconverting to base units7:47
Significant figureshow many figures do you write6:24
Line of best fitthe line touches no point5:07
Relationships and the constantbraking twice as fast5:20
Linearising a graphyour measurements never change4:41
About me
My name is Thomas Schuurmans, and I have a passion for physics and for understanding why things work the way they do. What I want is to get young people asking “why is that?” and “how does that work?” more often, and to hand them the tools of physics in a way that is simple and that you can see.
I graduated in applied physics at Delft University of Technology in 2003. After that I spent more than twenty years outside education: first at TNO, the Dutch applied research institute, then at a design agency, and eventually founding Proportion Global, through which I work on innovation questions in Africa, Latin America and South Asia. That work resembles physics more than you would expect: don't start from a solution, first understand what is going on, try something, be wrong, and look again.
Since 2026 I have been teaching physics to both lower and upper secondary classes, and I started a master's at the University of Amsterdam for my full teaching qualification. That is where I learned that a secondary school pupil's real attention for new material lasts about seven minutes. And my own weakness happens to be telling too many side stories.
That is where the idea came from: videos that explain one topic sharply and visually, inside those seven minutes. I make them for my own pupils. Then I publish them, because good explanation should be within reach of anyone who needs it, wherever you live and whatever language you think and speak in.