Functions Summary

Trigonometry · Axiom Academy

SUMMARY Trigonometric Functions Let's review how six fundamental functions capture circular and angular relationships, enabling countless real-world applications. Primary Functions: Sine and cosine form the foundation, representing vertical and horizontal coordinates on the unit circle Derived Functions: Tangent, cotangent, secant, and cosecant are defined in terms of sine and cosine Reciprocal Pairs: Three pairs of reciprocal functions: sin/csc, cos/sec, tan/cot Periodicity: All six functions repeat in regular patterns, making them ideal for modeling cyclical phenomena Unit Circle: For angle θ at point (x, y) on the unit circle: cos θ = x and sin θ = y Right Triangle (SOH-CAH-TOA): Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent Equivalence: Both approaches give identical results; use unit circle for any angle, right triangle for acute angles in triangles Domain Flexibility: Unit circle extends trig functions beyond 0° to 90°, working for negative angles and angles beyond 360° Cosecant: csc θ = 1/sin θ — Undefined when sin θ = 0 (at 0°, 180°, 360°, ...) Secant: sec θ = 1/cos θ — Undefined when cos θ = 0 (at 90°, 270°, ...) Cotangent: cot θ = 1/tan θ = cos θ/sin θ — Undefined when sin θ = 0 Key Strategy: To find sec, csc, or cot, first find cos, sin, or tan, then take the reciprocal Evaluating All Six Functions: Example at θ = 5π/4 Locate the angle: 5π/4 radians = 225°, which is in Quadrant III (between 180° and 270°)

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