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Math · Trigonometry

SOHCAHTOA & standard angles

Apply sine, cosine, and tangent ratios in right triangles, and recall exact values at 30°, 45°, and 60° from memory — the standard angles are what let you skip the calculator entirely. Every question comes with a written walkthrough of exactly which ratio applied.

Practise sohcahtoa & standard angles nowSee sample questions
3
Difficulty tiers
30s
Target pace
53%
Average first-attempt score

What's covered

  • SOHCAHTOA — sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent — always relative to the angle being used.
  • Solving for a missing side — set up the correct ratio using the given angle and one known side, then solve for the unknown side.
  • Solving for a missing angle — use the inverse trig functions — arcsin, arccos, arctan — when two sides are known and the angle itself is missing.
  • Exact values at standard angles — recall sin, cos, and tan at 30°, 45°, and 60° without a calculator, using the 30-60-90 and 45-45-90 special right triangles.
  • Angles of elevation and depression — translate a real-world description into a right triangle, identifying which side is opposite the given angle.

Where students lose marks

Labeling opposite and adjacent relative to the wrong angle

"Opposite" and "adjacent" only make sense relative to a specific angle — the same side can be opposite one angle and adjacent to another in the same triangle.

Confusing sine and cosine at standard angles

sin(30°) = 1/2 while cos(30°) = √3/2, and sin(60°) = √3/2 while cos(60°) = 1/2 — the two are mirror images of each other, and mixing them up under time pressure is the single most common error here.

Forgetting to take the inverse function when solving for an angle

Once a ratio like sin(x) = 0.5 is set up, x itself requires applying arcsin, not just reading off the ratio value as if it were the angle.

Three sample questions

Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.

Sample 01Foundation

In a right triangle, the angle is 30°, and the hypotenuse is 12. Find the length of the side opposite the 30° angle.

Sample 02Core

A right triangle has legs of length 8 and 15, with the right angle between them. Find the measure of the angle opposite the side of length 8, to the nearest degree.

Sample 03Advanced

A surveyor stands 50 meters from the base of a building and measures the angle of elevation to the top of the building as 40°. From the same spot, she measures the angle of elevation to a flagpole on top of the building as 48°. Find the height of the flagpole itself, not including the building, to the nearest tenth of a meter.

How to practise this

SOHCAHTOA problems are quick once the ratio and the standard-angle values are automatic — the setup is naming which side and angle you actually have, not the trig itself.

Label opposite, adjacent, and hypotenuse relative to the target angle every time

Before writing a ratio, mark which side is opposite and which is adjacent to the specific angle in question — don't assume it carries over from a different angle in the same triangle. Untimed practice is where that habit sticks.

Then 30 seconds a question

Move to timed sessions once ratio setup and standard-angle recall are instant. Multi-step problems like angles of elevation run a bit longer.

Fold into a mock

Pair this with Law of Sines and Law of Cosines in a mock — SOHCAHTOA is the right-triangle special case both laws generalize beyond.

Stop mislabeling the sides.
Start marking them every time.

  • A ratio-by-ratio walkthrough on every item
  • Timed or untimed sessions, any length you like
  • Tracked separately — see your accuracy and pace on this subtopic over time
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