T-Ratios for Angle of 30°

In ΔPMO and ΔP′MO,

OM = OM (Common)

∠PMO = ∠P′MO (Each = 90°)

 

PM = P′M (Construction)

∴ ΔPMO ≅ ΔP′MO

∴ ∠OPM = ∠OP′M = 60°

∴ OPP′ is an equilateral triangle

∴ OP = OP′

Let PM = a units

PP′ = PM + MP′ = 2a units

∴ OP = OP′ = PP′ = 2a units

In right triangle PMO,

OP2 = PM2 + OM2 (Pythagoras Theorem)

(2a)2 = a2 + OM2

∴ OM2 = 3a2

OM = √3a units

sin 30° = 1/2

cos 30° = √3/2

tan 30° = 1/√3

cosec 30° = 2

sec 30° = 2/√3

cot 30° = √3