SOHCAHTOA
Opposite and adjacent are not properties of a side. They are properties of a side relative to an angle - and nearly every SOHCAHTOA mistake is that sentence not having landed.
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SOHCAHTOA is a good mnemonic for a fact people usually already half-know, and it does nothing about the thing that actually goes wrong. The error is almost never the letters. It is that "opposite" and "adjacent" were treated as labels painted on the triangle, when they depend entirely on which angle you are standing at.
Change the triangle below, then switch the marked angle from A to B and watch the labels swap without a single line moving.
Switch the marked angle. The triangle does not move, but opposite and adjacent trade places - which is why those two words always need an angle attached.
Name the sides from the angle, not from the page
The hypotenuse is the easy one: always the longest side, always the one across from the right angle, and it never changes no matter which acute angle you are working from.
The other two do change:
- the opposite side is the one that does not touch your angle;
- the adjacent side is the leg that does.
That is the whole naming rule, and it is why the figure has an A/B toggle. In a 3-4-5 triangle, the side of length 4 is opposite one angle and adjacent to the other. Asking whether 4 "is the opposite" is a question with no answer until an angle is named.
The three ratios
With the sides named, the mnemonic finally means something:
| ratio | is | mnemonic |
|---|---|---|
| sine | opposite over hypotenuse | SOH |
| cosine | adjacent over hypotenuse | CAH |
| tangent | opposite over adjacent | TOA |
Two of them use the hypotenuse; tangent is the odd one out and uses neither. That is often the fastest way to choose: if the hypotenuse is nowhere in the question, you want tangent.
Why size does not matter
Here is the fact that makes trigonometry possible at all. Double every side of a right triangle and the angles are unchanged - and so is every ratio, because the doubling cancels:
(2*3)/(2*5) = 3/5
So sin of an angle is a property of the angle, not of any particular triangle. That is what lets a table of sines exist, and why a calculator can answer without being told the size of anything.
Move the two sliders in the figure while keeping their ratio the same - 3 and 4, then 6 and 8 - and watch the three readouts stay put.
Picking the right ratio
The method is always the same three steps.
- Mark your angle, then label the three sides from it.
- See which two sides the question mentions - one you know, one you want.
- Pick the ratio built from those two.
A ladder 10 metres long leans at 65° to the ground; how high does it reach? The ladder is the hypotenuse, the height is opposite the angle, so opposite and hypotenuse means sine: sin(65°) = h/10, and h = 10sin(65°).
Nothing about that reasoning involved remembering a formula for ladders.
Try one
A right triangle has legs 6 and 8, so its hypotenuse is 10. What is the sine of the angle opposite the side of length 6?
If you answered 6/8, the hypotenuse got mixed up with a leg - which is the one mistake the word "hypotenuse" exists to prevent.
Where this goes
Two directions from here. Some angles give exact ratios with no calculator at all - the 30-60-90 and 45-45-90 triangles - and those are worth knowing cold. Beyond that, the unit circle takes the same three ratios and frees them from the triangle entirely, so they keep meaning something past 90 degrees.
Common questions
- What does SOHCAHTOA stand for?
- Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent. It is a memory aid for the three ratios in a right triangle, and it only works once you know which side counts as opposite and which as adjacent - which depends on the angle you are working from.
- Which side is the opposite and which is the adjacent?
- Pick your angle first. The hypotenuse is always the longest side, across from the right angle, and never changes. Of the two remaining sides, the one that does not touch your angle is the opposite; the one that does is the adjacent. Switch to the other acute angle and those two swap.
- How do I know whether to use sin, cos or tan?
- Label the three sides relative to your angle, then see which two the question involves. If it mentions the opposite and the hypotenuse, that is sine; the adjacent and the hypotenuse, cosine; the two legs with no hypotenuse in sight, tangent. Choosing the ratio is a matter of what you have and what you want, not of the shape.
- Why does the ratio not change when the triangle gets bigger?
- Because scaling a triangle multiplies every side by the same factor, so any ratio of two sides is unchanged - the factor cancels top and bottom. Two right triangles with the same angles are similar, so sin of 30 degrees is the same number in a triangle the size of a coin and one the size of a field.
- Does SOHCAHTOA work on triangles without a right angle?
- No. All three ratios are defined from a right triangle, so a triangle with no right angle needs different tools - the law of sines or the law of cosines. A common workaround is to drop a perpendicular and split the triangle into two right ones, which is in fact how both laws are derived.
Now do it yourself, properly
Reading someone else's working is not the same as being able to do it. The Coddy math course puts you on a board that checks every move you make, so you find out where you actually stand.
