SOH-CAH-TOA: Sin, Cos, Tan from First Principles

Three ratios, one angle, every right-triangle problem. The visual lesson that turns sin/cos/tan from a memorization headache into a 5-second decision.

8 min TEKS 9A,9B Geometría

Why we need three ratios in the first place

Imagine you're standing on the ground looking up at the top of a building. You know how far you are from the base, and you can measure the angle you're looking up at. You want to know how tall the building is — without climbing up there with a tape measure.

That's the kind of problem the Texas Geometry CBE asks you to solve over and over again. To do it, you need a way to translate "angle + one side" into "the missing side". That translator has a name: trigonometry. And in a right triangle, it boils down to three ratios — sine, cosine, and tangent.

Big idea

For any right triangle, the ratio between two sides depends only on one of the non-right angles. Memorize three ratios — and you can find any side or angle from any pair of facts.

The anatomy of a right triangle

Before we name any ratios, let's name the parts. Pick one of the two non-right angles — call it θ (theta). Now look at the three sides relative to that angle:

Three sides, one labeled angle. Remember: opposite/adjacent always depend on which angle you're looking at.

Hypotenuse: The longest side, always across from the right angle. It never changes label — same regardless of which non-right angle you focus on.
Opposite: The side across from your chosen angle θ. Switch angles and this label switches sides.
Adjacent: The side next to θ that is not the hypotenuse. Switch angles and this also switches.

Watch out

"Opposite" and "adjacent" are not properties of a side — they're properties of a side relative to the angle you're working with. Pick the angle first, then label. The single most common mistake on the CBE is using the wrong pair.

The three ratios — SOH-CAH-TOA

Now the punchline. For a fixed angle θ, no matter how big or small you draw the right triangle, three ratios stay constant:

sin θ = opposite / hypotenuse cos θ = adjacent / hypotenuse tan θ = opposite / adjacent SOH — CAH — TOA

The acronym SOH-CAH-TOA just stitches the first letters together: Sin = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj. Tape it to your forehead before the test.

Aha moment

Notice how each ratio uses exactly two sides. If a problem gives you any one ratio plus any one side, you can solve for the other side. That's the entire trick.

Walkthrough — finding a missing side

Let's apply this to a real CBE-style problem. From a point on level ground 150 feet from the base of a building, you measure the angle of elevation to the top of the building as 32°. How tall is the building?

The 150 ft is adjacent to the 32° angle. The unknown height h is opposite. We never see the hypotenuse — so we want a ratio that uses only adjacent and opposite.

Step 1 — identify what you have and what you want. You have the adjacent side (150 ft) and the angle (32°). You want the opposite side (h).

Step 2 — pick the ratio that uses just those parts. tan is the only one that does: opposite / adjacent. So:

tan 32° = h / 150 h = 150 × tan 32° h = 150 × 0.625 = 93.75 ft

So the building is about 94 feet tall. The whole calculation took three lines once we picked tan correctly.

The decision tree

When you see a right-triangle word problem, ask: which two of {opposite, adjacent, hypotenuse} are involved? Opposite + adjacent → tan. Opposite + hypotenuse → sin. Adjacent + hypotenuse → cos. Pick before you write anything down.

Check yourself

Quick check #1

In a right triangle, sin(θ) is defined as:

Quick check #2

From the top of a 50-foot tower, the angle of depression to a point on the ground is 30°. How far is the point from the base? (Use tan 30° ≈ 0.577.)

Try it yourself

Now that you know the method, here's the kind of problem you'll see on the CBE. Click an answer to check your work — the explanation appears below.

Ladder against a wall

A 24-foot ladder leans against a wall, making a 65° angle with the level ground. To the nearest tenth of a foot, how high up the wall does the ladder reach? (sin 65° ≈ 0.906, cos 65° ≈ 0.423)

Wheelchair ramp angle

A wheelchair ramp must rise 18 inches over a horizontal run of 216 inches. To the nearest tenth of a degree, what is the angle of inclination of the ramp? (Use the inverse tangent: tan⁻¹.)

Where this goes next

SOH-CAH-TOA handles every right-triangle problem on the CBE that gives you an angle and one side. There are two natural extensions you'll see in upcoming lessons:

Inverse trig (sin⁻¹, cos⁻¹, tan⁻¹) — used when you know two sides and need to find the angle. Same three ratios, run backwards.
Special right triangles (30-60-90 and 45-45-90) — exact ratios you can write down without a calculator. The CBE loves these.

For now, the win condition is the same as for your CBE: given any right-triangle setup, you can pick the right ratio in under five seconds. That's it. That's the lesson.