Trigonometry

SOH CAH TOA

In a right-angled triangle, the three trigonometric ratios relate angles to side ratios:

SOH: sin θ = Opposite / Hypotenuse

CAH: cos θ = Adjacent / Hypotenuse

TOA: tan θ = Opposite / Adjacent

Identifying the Sides

Label sides relative to the angle θ:

• Hypotenuse (H): always opposite the right angle — the longest side
• Opposite (O): the side opposite to angle θ
• Adjacent (A): the side next to angle θ (not the hypotenuse)

Finding a Missing Side

Hypotenuse = 10cm, angle = 30°, find the opposite:

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

Finding a Missing Angle

tan θ = opp/adj = 7/7 = 1 → θ = tan⁻¹(1) = 45°

Use inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹) to find unknown angles.

Key Trig Values

θ	sin θ	cos θ	tan θ
0°	0	1	0
30°	0.5	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	0.5	√3
90°	1	0	undefined

Which Ratio to Use?

Given	Find	Use
H and θ	O	sin
H and θ	A	cos
A and θ	O	tan