#Mastering Sine and Cosine Graphs: Your Ultimate Guide

Hey there, future AP Pre-Calculus master! Let's dive into the world of sine and cosine graphs. Think of this as your backstage pass to understanding how these functions work, straight from the unit circle to the coordinate plane. Let's get started!

#The Unit Circle: Your Foundation

Before we graph, let's revisit the star of the show: the unit circle. It's a circle with a radius of 1, centered at the origin. It's where all the magic of sine and cosine begins.

Image courtesy of Remind.

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• The cosine of an angle is the x-coordinate of a point on the unit circle.
• As you move counterclockwise:
  • From 0 to π radians, cosine values decrease (1 to -1).
  • From π to 2π radians, cosine values increase (-1 to 1).
• Cosine values range from -1 to 1.
  • Maximum (1) at 0 radians (1,0).
  • Minimum (-1) at π radians (-1,0).
  • Zero at π/2 and 3π/2 radians.

#Sines: The Y-Factor

• The sine of an angle is the y-coordinate of a point on the unit circle.
• As you move counterclockwise:
  • From 0 to π/2 radians, sine values increase (0 to 1).
  • From π/2 to 3π/2 radians, sine values decrease (1 to -1).
  • From 3π/2 to 2π radians, sine values increase (-1 to 0).
• Sine values range from -1 to 1.
  • Maximum (1) at π/2 radians (0,1).
  • Minimum (-1) at 3π/2 radians (0,-1).
  • Zero at 0...