Trigonometric Ratios of Special Angles

Master 0°, 30°, 45°, 60°, and 90° through geometric derivations

INTERMEDIATE LEVEL
sin²θ + cos²θ = 1
The foundation of all trigonometric relationships
Welcome to trigonometric ratios of special angles! We'll derive exact values for 0°, 30°, 45°, 60°, and 90° using geometric constructions.
📐 45° Angle (π/4)

Derived from an isosceles right triangle where both legs are equal. This creates perfect symmetry between sine and cosine.

sin 45° = cos 45° = 1/√2
Perfect symmetry in the unit circle
🔺 30° & 60° Angles

Derived from an equilateral triangle by drawing an altitude. This creates a 30-60-90 triangle with sides in ratio 1:√3:2.

sin 30° = 1/2, cos 60° = 1/2
Complementary angle relationships
♾️ Limiting Cases

0° and 90° represent the extremes where the triangle collapses to a line, giving us boundary values for all trigonometric functions.

sin 0° = 0, cos 0° = 1
Boundary behavior analysis
🎯 Practical Applications

These exact values appear everywhere: engineering calculations, physics problems, and geometric constructions. Memorizing them saves countless calculations!

Essential for exact solutions
Real-world problem solving

Complete Table of Special Angle Values

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