T-Ratios for Angle of 60°

In ΔPMO and ΔPMM′,

PM = PM (Common)

∠PMO = ∠PMM′ (Each = 90°)

 

OM = MM′ (Construction)

∴ ΔPMO ≅ ΔPMM′

∴ ∠POM = ∠PM′M = 60°

∴ ΔPOM′ is an equilateral triangle.

∴ OP = PM′ = OM′ = 2a units

In right ΔPMO,

OP2 = PM2 + OM2 (Pythagorus Theorem)

(2a)2 = PM2 + a2

PM2 = 3a2

PM = √3a units

sin 60° = √3/2

cos 60° = 1/2

tan 60° = √3

cosec 60° = 2/√3

sec 60° = 2

cot 60° = 1/√3