Three basic ratios that connect a right triangle's angles to its side lengths.
The Mnemonic
Three Ratios, One Angle's Perspective
In a right triangle, once you pick one of the two non-right angles (call it θ), the three sides get relabeled from THAT angle's point of view: the hypotenuse is always the side opposite the right angle (this never changes), the opposite side is directly across from θ, and the adjacent side is the other leg touching θ.
The three basic trig ratios compare pairs of these sides: sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), and tangent (opposite/adjacent).
From angle θ's viewpoint: opposite is across from it, adjacent is next to it, hypotenuse is always across from the right angle
💡 Memory Trick
"SOH-CAH-TOA" — Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Each three-letter chunk spells out the ratio directly: the first letter is the trig function, and the next two letters give numerator-over-denominator in order — S-O-H means sin = opposite over hypotenuse.
Why It Works
Why the Ratio Stays the Same
Here's the genuinely useful fact underneath SOH-CAH-TOA: for a FIXED angle θ, the ratio of opposite/hypotenuse (or any of the three ratios) is always exactly the same number, no matter how big or small the actual triangle is. A tiny right triangle with a 30° angle and a massive right triangle with the same 30° angle will have identical sine, cosine, and tangent values.
This works because any two right triangles sharing the same acute angle are similar (matching angles), and similar triangles always have proportional sides — so the ratio between any two sides stays constant even as the actual side lengths scale up or down. That's exactly what lets a single number (like sin 30° = 0.5) apply to every 30-60-90 triangle that has ever existed or ever will.
Using It In A Proof
Choosing the Right Ratio
Solving a right-triangle problem with trig comes down to correctly identifying which two sides are involved relative to the angle you're using.
1
Identify the angle you're working from
Pick the angle θ (given or the one you need to find) — every label (opposite/adjacent) depends entirely on which angle you chose.
2
Label the two sides involved
Figure out which two sides the problem gives you or asks for, and determine whether they're opposite, adjacent, or the hypotenuse relative to θ.
3
Match to the correct ratio
Opposite & hypotenuse → sine. Adjacent & hypotenuse → cosine. Opposite & adjacent → tangent. Set up the equation and solve.
Full Worked Example
Finding a Missing Side Using Tangent
Given: A right triangle has an angle of 35°, and the side adjacent to it measures 20. Find: the length of the opposite side.
1
Identify the two sides involved
The problem gives the adjacent side and asks for the opposite side — that combination matches tangent (TOA).
2
Set up the equation
tan(35°) = opposite / adjacent = opposite / 20.
3
Solve for the opposite side
opposite = 20 × tan(35°) ≈ 20 × 0.7002 ≈ 14.0.
Always multiply by tan(θ) when solving for a numerator side — the unknown side is isolated by moving the denominator (20) across, not by dividing.
🎯 Quick Worked Example
A right triangle has a hypotenuse of 10 and an angle of 40°. Find the side opposite that angle.
1
Identify the sides involved. The problem gives the hypotenuse and asks for the opposite side — that's sine (SOH).
A fast way to double-check which ratio you need: hypotenuse is ALWAYS one side of sine or cosine, never tangent. If a problem doesn't mention the hypotenuse at all, it must be a tangent problem — this quick filter avoids the most common mix-up between the three ratios.
⚠️ Most Common Trig Ratios Mistakes
Trap 1 — Mislabeling opposite vs. adjacent: These labels depend entirely on which angle you're using as θ — the same side can be "opposite" from one angle's perspective and "adjacent" from the other non-right angle's perspective in the same triangle.
Trap 2 — Dividing instead of multiplying when solving: When the unknown side is in the numerator (like opposite in tan θ = opposite/adjacent), you multiply both sides by the denominator — dividing instead is a common algebra slip that produces a much smaller, incorrect answer.
✓ Quick Self-Test
1) What does SOH-CAH-TOA stand for? 2) Why does the ratio stay the same for a fixed angle regardless of triangle size? 3) A right triangle has an angle of 50° and a hypotenuse of 8 — which ratio finds the adjacent side? 4) If a problem never mentions the hypotenuse, which ratio must be used? 5) Set up (don't solve) the equation for finding the opposite side if you know the adjacent side is 15 and the angle is 28°.