Trigonometry: SOH-CAH-TOA

In right-angled triangles, trigonometry links the sides and angles. The key to IGCSE 0580 is remembering the three ratios correctly: SOH-CAH-TOA.

The three ratios

Relative to the angle (θ): the hypotenuse (longest side, opposite the right angle), the opposite (across from the angle) and the adjacent (next to the angle).

• SOH → sin θ = Opposite / Hypotenuse
• CAH → cos θ = Adjacent / Hypotenuse
• TOA → tan θ = Opposite / Adjacent

Finding a side

Angle 30°, hypotenuse 10 cm. Find the opposite side.

Opposite + hypotenuse → use sin:

sin 30° = opp / 10 → opp = 10 × sin 30° = 10 × 0.5 = 5 cm

Finding an angle

If two sides are known, use the inverse function (sin⁻¹, cos⁻¹, tan⁻¹).

Opposite = 4, adjacent = 3 → find the angle.

tan θ = 4/3 → θ = tan⁻¹(4/3) = 53.1°

The most common mistake

Make sure your calculator is in DEG (degrees) mode — in RAD mode the answer is wrong. Also decide which ratio you need first: whichever two sides you have (O/A/H) tells you which ratio to choose.

Exam tip

Draw the triangle, label the sides O / A / H, then pick the right ratio from SOH-CAH-TOA. These three steps score full marks in 0580. The sine/cosine rule (for non-right-angled triangles) is a separate topic — don't mix it up with these ratios.