T-Ratios for Angle of 30°

Let a ray OA start from OX and rotate in the anti clockwise direction and make an angle of 30° with x-axis.

Take any point P on OA. Draw PM ⊥ OX and produce PM to P′ such that PM = P′M. Join OP′

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

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

$$ \sin 30° = \frac{1}{2} $$

$$ \cos 30° = \frac{\sqrt{3}}{2} $$

$$ \tan 30° = \frac{1}{\sqrt{3}} $$

$$ \csc 30° = 2 $$

$$ \sec 30° = \frac{2}{\sqrt{3}} $$

$$ \cot 30° = \sqrt{3} $$