Trigonometry (SOHCAHTOA): Complete Guide
Trigonometry lets you find missing sides and angles in right-angled triangles using the ratios sine, cosine and tangent. The memory aid is SOHCAHTOA.
SOHCAHTOA
SOH CAH TOA
- SOH: sinθ = Opposite ÷ Hypotenuse
- CAH: cosθ = Adjacent ÷ Hypotenuse
- TOA: tanθ = Opposite ÷ Adjacent
Label the sides relative to the angle θ: the hypotenuse is opposite the right angle, the opposite is across from θ, and the adjacent is next to θ.
Worked example: finding a side
Find the opposite side when θ = 30° and the hypotenuse = 10 cm
Step 1: Choose the ratio with O and H → sin
Step 2: sin 30° = O ÷ 10
Step 3: O = 10 × sin 30° = 10 × 0.5 = 5 cm
Worked example: finding an angle
Find θ when the opposite = 4 cm and adjacent = 3 cm
Step 1: Choose the ratio with O and A → tan
Step 2: tanθ = 4 ÷ 3
Step 3: θ = tan⁻¹(4 ÷ 3) = 53.1° (1 dp)
Common mistakes
- Mislabelling opposite vs adjacent
- Calculator in the wrong mode — use degrees for GCSE
- Forgetting the inverse (sin⁻¹, cos⁻¹, tan⁻¹) when finding an angle
Frequently asked questions
What does SOHCAHTOA mean?
It is a memory aid: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.
How do I find a missing angle?
Pick the ratio that uses the two sides you know, then apply the inverse function (sin-1, cos-1 or tan-1) to find the angle.
Should my calculator be in degrees or radians?
For GCSE trigonometry use degrees mode. Check the display shows DEG before working out angles.
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