The Three Trig Ratios
Right triangle trigonometry is built on three ratios that relate the angles of a right triangle to the lengths of its sides. Pick any acute angle (not the 90° angle) in the triangle and label it θ. Relative to that angle, the three sides have specific names.
The three sides relative to θ
The opposite side is the one directly across from θ. The adjacent side is the one touching θ that is not the hypotenuse. The hypotenuse is the longest side — always opposite the right angle.
From those three sides come three ratios: sin(θ) = opposite / hypotenuse, cos(θ) = adjacent / hypotenuse, and tan(θ) = opposite / adjacent.
These ratios depend only on the angle, not on the size of the triangle. Two right triangles with the same acute angle will have the same sin, cos, and tan values even if one is ten times bigger than the other. That is why trigonometry works — the ratios encode the shape of the triangle independent of scale.
Once you know one acute angle and one side, these three ratios let you compute every other side and angle in the triangle.
SOH CAH TOA — The Mnemonic
SOH CAH TOA is the classic mnemonic that every trigonometry student learns first, and for good reason — it captures all three right triangle ratios in nine letters.
SOH: Sin = Opposite / Hypotenuse. CAH: Cos = Adjacent / Hypotenuse. TOA: Tan = Opposite / Adjacent.
How to apply it
Whenever you face a right triangle problem, the first move is always the same: identify the angle you are working with, label the three sides as opposite, adjacent, and hypotenuse relative to that angle, and then decide which of the three ratios to use based on which two sides are involved.
| Case | Sides Involved | Ratio to Use |
|---|---|---|
| 1 | Opposite and hypotenuse | sine |
| 2 | Adjacent and hypotenuse | cosine |
| 3 | Opposite and adjacent | tangent |
Some students remember SOH CAH TOA as a nonsense name (“Soh-Kah-Toh-Ah”), others use the phrase “Some Old Hippie Caught Another Hippie Tripping On Acid” — whichever sticks in your head is the right one. What matters is that the mnemonic gives you instant recall under pressure.
Finding a Missing Side
To find a missing side when you know one acute angle and one other side, follow these three steps.
- Label the sides
Label the sides as opposite, adjacent, and hypotenuse relative to the known angle.
- Pick the ratio
Pick the ratio (sin, cos, or tan) that uses the side you know and the side you want.
- Solve
Write the equation, plug in, and solve.
A right triangle has an acute angle of 35° and a hypotenuse of 20. Find the side opposite the 35° angle.
The ratio that uses opposite and hypotenuse is sine, so sin(35°) = opposite / 20.
Solving gives opposite = 20 · sin(35°) ≈ 20 · 0.5736 ≈ 11.47.
opposite ≈ 11.47A right triangle has an acute angle of 50° and an adjacent side of 8. Find the hypotenuse.
Adjacent and hypotenuse go with cosine: cos(50°) = 8 / hypotenuse.
So hypotenuse = 8 / cos(50°) ≈ 8 / 0.6428 ≈ 12.45.
hypotenuse ≈ 12.45In a right triangle, the angle is 25° and the adjacent side is 15. Find the opposite side.
Use tangent: tan(25°) = opposite / 15.
So opposite = 15 · tan(25°) ≈ 15 · 0.4663 ≈ 7.00.
opposite ≈ 7.00The pattern is always the same — pick the correct ratio, substitute, and solve.
Finding a Missing Angle with Inverse Trig
When you know two sides of a right triangle and want to find an angle, you use the inverse trig functions: sin⁻¹, cos⁻¹, and tan⁻¹ (also written arcsin, arccos, arctan). These functions take a ratio and return the angle whose sine, cosine, or tangent equals that ratio.
The method mirrors finding a side, but in reverse.
- Identify the sides
Identify which two sides you know relative to the angle you want.
- Pick the ratio
Pick the ratio that uses those two sides.
- Apply the inverse function
Apply the inverse function.
A right triangle has opposite = 7 and hypotenuse = 13. Find the angle.
Opposite and hypotenuse → sine, so sin(θ) = 7/13 ≈ 0.5385.
And θ = sin⁻¹(0.5385) ≈ 32.58°.
θ ≈ 32.58°Adjacent = 10, hypotenuse = 18.
Adjacent and hypotenuse → cosine, so cos(θ) = 10/18 ≈ 0.5556.
And θ = cos⁻¹(0.5556) ≈ 56.25°.
θ ≈ 56.25°Opposite = 6, adjacent = 9.
Opposite and adjacent → tangent, so tan(θ) = 6/9 = 0.6667.
And θ = tan⁻¹(0.6667) ≈ 33.69°.
θ ≈ 33.69°On your calculator, make sure it is set to degree mode (or radians, depending on what the problem asks) — this is the single most common source of wrong answers in inverse trig.
Special Right Triangles (30-60-90 and 45-45-90)
Two right triangles appear so often that their side ratios are worth memorizing — no calculator required.
The 45-45-90 triangle
The 45-45-90 triangle (also called an isosceles right triangle) has two 45° angles and one 90° angle. Its sides are in the ratio 1 : 1 : √2, with the two legs equal and the hypotenuse being √2 times the leg length. If each leg is x, the hypotenuse is x√2.
A 45-45-90 triangle with legs of length 5 has hypotenuse 5√2 ≈ 7.07.
hypotenuse = 5√2 ≈ 7.07The 30-60-90 triangle
The 30-60-90 triangle has angles of 30°, 60°, and 90°. Its sides are in the ratio 1 : √3 : 2, where 1 is opposite the 30° angle, √3 is opposite the 60° angle, and 2 is the hypotenuse (opposite 90°). If the shortest side is x, then the medium side is x√3 and the hypotenuse is 2x.
A 30-60-90 triangle with shortest side 4 has sides 4, 4√3 ≈ 6.93, and 8.
4, 4√3 ≈ 6.93, 8Because these ratios are exact, they produce exact answers with no calculator rounding. Standardized tests (SAT, ACT, AP) rely heavily on these two triangles, so memorizing them gives you a significant speed advantage.
Real-World Applications
Right triangle trigonometry is one of the most applied topics in mathematics.
| Field | How Trig Is Used |
|---|---|
| Surveying and construction | Use trig to measure distances and heights that cannot be reached directly. To find the height of a building, a surveyor measures the angle of elevation from a known distance and computes height = distance · tan(angle). |
| Navigation | Uses trig to convert bearings and distances into coordinates, whether for ships, planes, or hiking in the backcountry. |
| Physics | Uses trig constantly to break forces, velocities, and accelerations into perpendicular components — the horizontal component of a force at angle θ is F·cos(θ), and the vertical component is F·sin(θ). |
| Engineering | Uses trig for ramps, bridges, roof pitches, and anything involving angles. |
| Computer graphics and game development | Use trig for rotations, camera angles, and projectile motion. |
Even in everyday situations — figuring out the length of a ladder needed to reach a window, the steepness of a hiking trail, or the angle at which to aim a camera — the same SOH CAH TOA relationships apply.
The next time you need to set up a right triangle problem, snap a photo with Solver AI to see the correct ratio picked, the equation solved, and the answer verified step by step.