Value of tan 30°

tan 30° = 1/√3

Rationalised form: 1/√3 × √3/√3 = √3/3

Decimal: tan 30° ≈ 0.5774

Derivation from 30-60-90 triangle: In a 30-60-90 triangle with hypotenuse = 2: • Side opposite 30° = 1 • Side opposite 60° = √3 • Hypotenuse = 2

tan 30° = opposite/adjacent = 1/√3

Alternative using sin and cos: tan 30° = sin 30° / cos 30° = (1/2) / (√3/2) = (1/2) × (2/√3) = 1/√3

Angle → sin → cos → tan 0° → 0 → 1 → 0 30° → 1/2 → √3/2 → 1/√3 45° → 1/√2 → 1/√2 → 1 60° → √3/2 → 1/2 → √3 90° → 1 → 0 → undefined

Decimal values: 0° → 0 → 1 → 0 30° → 0.5 → 0.866 → 0.577 45° → 0.707 → 0.707 → 1 60° → 0.866 → 0.5 → 1.732 90° → 1 → 0 → —

Memory trick for sin: sin 0°=√0/2, sin 30°=√1/2, sin 45°=√2/2, sin 60°=√3/2, sin 90°=√4/2 → 0, 1/2, 1/√2, √3/2, 1 tan = sin/cos

Exact fraction: 1/√3 Rationalised: √3/3 Decimal: 0.57735...

Related values: • tan 60° = √3 ≈ 1.732 (reciprocal relationship: tan 60° = 1/tan 30°) • cot 30° = √3 (cot = 1/tan) • tan 30° = sin 30°/cos 30° = (1/2)/(√3/2) = 1/√3

Reciprocal identity: cot 30° = 1/tan 30° = √3

Pythagorean identity check: sin²30° + cos²30° = (1/2)² + (√3/2)² = 1/4 + 3/4 = 1 ✓

tan 30° is used in: • Finding heights using angle of elevation (30°) • Equilateral triangle calculations (each angle = 60°, half = 30°) • Trigonometric simplifications