Overview

This topic introduces the six trigonometric functions, their definitions, graphs, and key properties. You'll learn how to evaluate trig functions using the unit circle and right triangle ratios, and how to apply them in various contexts.

Key Concepts and Definitions

• Trig Functions: Sine (sin), Cosine (cos), Tangent (tan), Cosecant (csc), Secant (sec), Cotangent (cot)
• Right Triangle Definitions:
  • sin(θ) = \frac{\text{opposite}}{\text{hypotenuse}}
  • cos(θ) = \frac{\text{adjacent}}{\text{hypotenuse}}
  • tan(θ) = \frac{\text{opposite}}{\text{adjacent}}
• Unit Circle: A circle of radius 1 centered at the origin. Coordinates of points correspond to (cos(θ), sin(θ)).
• Radians: Most trig functions are defined in radians. π radians = 180°
• Common Angles: Know exact values for 0°, 30°, 45°, 60°, 90° (or 0, π/6, π/4, π/3, π/2)
• Graphs: Understand period, amplitude, phase shift, and vertical shift for sin(x), cos(x), and tan(x)

Step-by-Step Example

Problem: Evaluate sin(π/6) using the unit circle.

Step 1: Identify the angle on the unit circle:
π/6 = 30°

Step 2: From the unit circle, the coordinates at π/6 are (\sqrt{3}/2, 1/2)

Step 3: sin(π/6) is the y-coordinate:
sin(π/6) = 1/2

Answer: 1/2

Quick Tip