T-Ratios for Angle of 60°

Let a ray OA start from OX and rotate in anti-clockwise direction and make an angle of 60° with x-axis. Take any point P on OA.

Draw PM ⊥ OX. Produce OM to M′ such that OM = MM′. Join PM′.

Let OM = a units

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

$$ PM = \sqrt{3}a $$

$$ \sin 60° = \frac{\sqrt{3}}{2} $$

$$ \cos 60° = \frac{1}{2} $$

$$ \tan 60° = \sqrt{3} $$

$$ \csc 60° = \frac{2}{\sqrt{3}} $$

$$ \sec 60° = 2 $$

$$ \cot 60° = \frac{1}{\sqrt{3}} $$