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§ Trigonometry

Trigonometry (SOH CAH TOA)

§ Trigonometry

Trigonometry (SOH CAH TOA)

CCSS.HSG.SRT3 min read

SOH CAH TOA is a mnemonic device for remembering the three fundamental trigonometric ratios in right triangles: sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent. These ratios appear throughout Year 9 mathematics and form the foundation for GCSE trigonometry. The acronym stands for Sine-Opposite-Hypotenuse, Cosine-Adjacent-Hypotenuse, and Tangent-Opposite-Adjacent.

§ 01

Why it matters

Trigonometry underpins countless real-world calculations, from architects determining roof angles to engineers designing bridges. Surveyors use trigonometry to measure inaccessible distances—a building's height can be calculated by measuring its shadow and the sun's angle. In navigation, pilots use trigonometric ratios to calculate flight paths, whilst sailors determine their position using celestial angles. The construction industry relies on these ratios for everything from staircase gradients (typically 35° to 40°) to ensuring walls are perfectly vertical. Video game programmers use trigonometry for realistic movement and collision detection. At GCSE level, these skills directly support further mathematics, physics, and engineering studies, where students encounter vectors, waves, and rotational motion. Even seemingly simple tasks like positioning a ladder safely (at a 75° angle to avoid slipping) demonstrate trigonometry's practical importance in everyday safety calculations.

§ 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 tan(60°)?

Answer: √3

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

In a right triangle, the hypotenuse is 10 and angle A = 36.9°. Find the opposite side.

Answer: 6

  1. Draw and label the triangle → Right angle at C, angle A = 36.9°, hypotenuse = 10 — Always start by labelling the sides relative to the given angle: opposite, adjacent, hypotenuse.
  2. Choose the right ratio using SOH CAH TOA → We know: hypotenuse. We want: opposite → use SOH (sin) — We have the hypotenuse and want the opposite side, so we need sin = opposite/hypotenuse.
  3. Write the equation → sin(36.9°) = opposite / 10 — Substitute the known values into the formula.
  4. Solve for the opposite side → opposite = 10 × sin(36.9°) = 10 × 0.6 = 6 — Multiply both sides by the hypotenuse to isolate the opposite side.
  5. Verify with the Pythagorean theorem → 6² + 8² = 36 + 64 = 100 = 10² ✓ — a² + b² = c² confirms our answer is correct.
Medium§ 03

In a right triangle with opposite = 5 and adjacent = 12, find angle A.

Answer: 22.6°

  1. Identify the known sides → opposite = 5, adjacent = 12 — 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) = 512 = 0.4167 — Substitute the known side lengths into the tangent ratio.
  4. Use the inverse function to find the angle → A = tan⁻¹(0.4167) = 22.6° — Press tan⁻¹ (or arctan) on your calculator to go from ratio back to angle.
  5. Sanity check → A = 22.6° (between 0° and 90° ✓) — The answer must be between 0° and 90° for a right triangle. 22.6° is reasonable since opposite < adjacent.
§ 04

Common mistakes

  • Confusing which side is opposite or adjacent, leading to tan(30°) = √3/3 being calculated as √3 instead of 1/√3
  • Using degrees instead of the inverse function, writing sin(0.6) = 36.9° instead of sin⁻¹(0.6) = 36.9°
  • Mixing up the ratios, calculating sin(45°) as 1/√2 instead of √2/2, or writing cos(60°) = √3/2 instead of 1/2
  • Forgetting to convert between radians and degrees, giving tan(π/4) = 57.3 instead of tan(45°) = 1
§ 05

Frequently asked questions

What does SOH CAH TOA stand for?
SOH CAH TOA represents three trigonometric ratios: SOH means Sine equals Opposite over Hypotenuse, CAH means Cosine equals Adjacent over Hypotenuse, and TOA means Tangent equals Opposite over Adjacent. This mnemonic helps recall which ratio connects which sides in right triangles.
How do I remember which side is opposite and which is adjacent?
The opposite side sits directly across from the angle being measured, whilst the adjacent side touches the angle but isn't the hypotenuse. The hypotenuse is always the longest side, opposite the right angle. Label your triangle carefully before applying SOH CAH TOA.
When do I use sin⁻¹, cos⁻¹, or tan⁻¹?
Use inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) when finding an angle from known side lengths. For example, if opposite = 3 and hypotenuse = 5, then the angle = sin⁻¹(3/5) = 36.9°. Regular sin, cos, tan give ratios from angles.
What are the exact values for 30°, 45°, and 60°?
At 30°: sin = 1/2, cos = √3/2, tan = 1/√3. At 45°: sin = √2/2, cos = √2/2, tan = 1. At 60°: sin = √3/2, cos = 1/2, tan = √3. These special angles appear frequently in GCSE questions and should be memorised.
Why doesn't SOH CAH TOA work for obtuse triangles?
SOH CAH TOA only applies to right triangles because it relies on the concept of opposite, adjacent, and hypotenuse sides. Obtuse triangles lack a right angle, so these side relationships don't exist. For obtuse triangles, use the sine rule or cosine rule instead.
§ 06

See also

HypotenuseFunctionRatioRight angle
§ 06

Where to next?

Prerequisites
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