Skip to content
MathAnvil
§ Trigonometry

Trigonometry (SOH CAH TOA)

§ Trigonometry

Trigonometry (SOH CAH TOA)

CCSS.HSG.SRT3 min read

Trigonometry uses ratios to relate the angles and sides of right triangles through three fundamental functions: sine, cosine, and tangent. The mnemonic SOH CAH TOA helps recall these relationships: SOH means sine equals opposite over hypotenuse, CAH means cosine equals adjacent over hypotenuse, and TOA means tangent equals opposite over adjacent. These ratios form the foundation for solving problems involving right triangles and appear in CCSS.HSG.SRT standards for geometric problem-solving.

§ 01

Why it matters

Trigonometry appears throughout engineering, physics, and construction where precise angle and distance calculations are essential. Architects use these ratios to determine roof slopes, calculating that a 30° roof angle requires a rise of 6 feet for every 10.4 feet of horizontal distance. Surveyors apply trigonometry to measure inaccessible distances, such as finding a building's height by measuring the angle of elevation and distance from the base. In navigation, pilots use trigonometric functions to calculate flight paths and distances. Video game programmers rely on sine and cosine functions to create realistic motion and rotation effects. Medical imaging technology, including MRI and CT scans, uses advanced trigonometric principles to reconstruct 3D images from 2D data, demonstrating how these basic ratios scale to complex applications.

§ 02

How to solve trigonometry (soh cah toa)

Trigonometry (SOH CAH TOA)

  • sin(A) = Opposite / Hypotenuse (SOH).
  • cos(A) = Adjacent / Hypotenuse (CAH).
  • tan(A) = Opposite / Adjacent (TOA).
  • To find an angle: use inverse functions (sin⁻¹, cos⁻¹, tan⁻¹).

Example: sin(30°) = 12, cos(60°) = 12.

§ 03

Worked examples

Beginner§ 01

What is cos(30°)?

Answer: √32

  1. Recall the mnemonic SOH CAH TOA → CAH: cos = adjacent/hypotenuse — SOH = Sine-Opposite-Hypotenuse, CAH = Cosine-Adjacent-Hypotenuse, TOA = Tangent-Opposite-Adjacent.
  2. Identify what cos means → cos = adjacent/hypotenuse — We need cos(30°), which is the ratio adjacent/hypotenuse.
  3. Look up the standard value for 30° → cos(30°) = √32 — The angles 30°, 45° and 60° have exact values you should memorise.
Easy§ 02

In a right triangle with opposite = 3 and adjacent = 4, find angle A.

Answer: 36.9°

  1. Identify the known sides → opposite = 3, adjacent = 4 — We know two sides: the opposite and the adjacent (relative to angle A).
  2. Choose the right ratio using SOH CAH TOA → We know: opposite + adjacent → use TOA (tan) — We have opposite and adjacent, so we use tan = opposite/adjacent.
  3. Write the equation → tan(A) = 34 = 0.75 — Substitute the known side lengths into the tangent ratio.
  4. Use the inverse function to find the angle → A = tan⁻¹(0.75) = 36.9° — Press tan⁻¹ (or arctan) on your calculator to go from ratio back to angle.
  5. Sanity check → A = 36.9° (between 0° and 90° ✓) — The answer must be between 0° and 90° for a right triangle. 36.9° is reasonable since opposite < adjacent.
Medium§ 03

In a right triangle with opposite = 8 and adjacent = 15, find angle A.

Answer: 28.1°

  1. Identify the known sides → opposite = 8, adjacent = 15 — We know two sides: the opposite and the adjacent (relative to angle A).
  2. Choose the right ratio using SOH CAH TOA → We know: opposite + adjacent → use TOA (tan) — We have opposite and adjacent, so we use tan = opposite/adjacent.
  3. Write the equation → tan(A) = 815 = 0.5333 — Substitute the known side lengths into the tangent ratio.
  4. Use the inverse function to find the angle → A = tan⁻¹(0.5333) = 28.1° — Press tan⁻¹ (or arctan) on your calculator to go from ratio back to angle.
  5. Sanity check → A = 28.1° (between 0° and 90° ✓) — The answer must be between 0° and 90° for a right triangle. 28.1° is reasonable since opposite < adjacent.
§ 04

Common mistakes

  • Confusing which side is opposite or adjacent, leading to tan(30°) = √3 instead of 1/√3 when the sides are mislabeled
  • Using degrees instead of radians on calculators, producing sin(30) = -0.988 instead of sin(30°) = 0.5
  • Forgetting to use inverse functions when finding angles, writing tan(A) = 0.75 = 36.9° instead of A = tan⁻¹(0.75) = 36.9°
§ 05

Frequently asked questions

What does SOH CAH TOA stand for?
SOH CAH TOA is a mnemonic where SOH means Sine = Opposite/Hypotenuse, CAH means Cosine = Adjacent/Hypotenuse, and TOA means Tangent = Opposite/Adjacent. Each letter represents the first letter of the trigonometric function and the sides of a right triangle it relates.
How do you remember which side is opposite and which is adjacent?
The opposite side is across from the angle in question, while the adjacent side touches the angle. The hypotenuse is always the longest side, opposite the right angle. If working with angle A, the opposite side is across from A, and the adjacent side connects to A.
When should you use inverse trigonometric functions?
Use inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) when finding an angle from known side lengths. If you know the ratio and need the angle, apply the inverse. For example, if tan(A) = 0.75, then A = tan⁻¹(0.75) = 36.9°.
What are the exact values for 30°, 45°, and 60° angles?
For 30°: sin = 1/2, cos = √3/2, tan = 1/√3. For 45°: sin = √2/2, cos = √2/2, tan = 1. For 60°: sin = √3/2, cos = 1/2, tan = √3. These special angles appear frequently in problems and should be memorized.
Why do trigonometric ratios only work for right triangles?
SOH CAH TOA definitions require a right angle to establish the hypotenuse and create consistent opposite/adjacent relationships. For non-right triangles, mathematicians use the Law of Sines and Law of Cosines, which extend trigonometric principles beyond the basic SOH CAH TOA framework.
§ 06

See also

HypotenuseBaseFunctionRight angleRatio
§ 06

Where to next?

Prerequisites
—
Share this article