Overview
This lecture covers the definitions of the six trigonometric functions using right triangles and unit circles, their signs in different quadrants, a mnemonic for remembering them, and how to calculate these functions for a point in any quadrant.
Trig Functions in Right Triangles
- In a right triangle with sides x (adjacent), y (opposite), and hypotenuse r, sine θ = y/r.
- On the unit circle (r = 1), sine θ = y/1 = y.
- Cosine θ = x/r.
- Tangent θ = y/x.
- Cosecant θ = r/y (reciprocal of sine).
- Secant θ = r/x (reciprocal of cosine).
- Cotangent θ = x/y (reciprocal of tangent).
Signs of Trig Functions in Quadrants
- Quadrant I: All trig functions are positive.
- Quadrant II: Sine is positive; others are negative.
- Quadrant III: Tangent is positive; others are negative.
- Quadrant IV: Cosine is positive; others are negative.
- "All Students Take Calculus" helps remember which function is positive in each quadrant.
Example 1: Point (-5, 12)
- x = -5, y = 12, r = 13.
- Sine θ = 12/13.
- Cosine θ = -5/13.
- Tangent θ = -12/5.
- Cosecant θ = 13/12.
- Secant θ = -13/5.
- Cotangent θ = -5/12.
Example 2: Point (-8, -15)
- x = -8, y = -15, r = 17.
- Sine θ = -15/17.
- Cosine θ = -8/17.
- Tangent θ = 15/8.
- Cosecant θ = -17/15.
- Secant θ = -17/8.
- Cotangent θ = 8/15.
Example 3: Point (2, -4)
- x = 2, y = -4, r = 2√5 (from a² + b² = c²).
- Sine θ = -2√5/5 (after rationalizing -4/(2√5)).
- Cosine θ = √5/5 (after rationalizing 2/(2√5)).
- Tangent θ = -2.
- Cosecant θ = -√5/2.
- Secant θ = √5.
- Cotangent θ = -1/2.
Key Terms & Definitions
- Hypotenuse (r) — Longest side of a right triangle, opposite the right angle.
- Adjacent — Side next to the angle θ.
- Opposite — Side opposite the angle θ.
- Rationalizing — Process of removing a square root from the denominator of a fraction.
- Unit Circle — Circle with radius 1 centered at the origin.
Action Items / Next Steps
- Memorize trig function definitions and their signs in each quadrant.
- Practice finding all six trig functions from any given point (x, y).