10.2 Trigonometric Graph Properties


2026 📋 Syllabus Objectives

By the end of these notes, you will be able to:

  • Understand what amplitude and period mean for a trigonometric function
  • Use amplitude and period to describe and sketch trigonometric graphs
  • Understand the relationship between related trigonometric graphs (for example, how y=sinxy = \sin x and y=3sin2xy = 3\sin 2x are connected)
  • Work with periods measured in both degrees and radians

1. What Are Trigonometric Graphs?

The three main trigonometric functions — sine, cosine, and tangent — all produce graphs with special, repeating shapes.

Think of a Ferris wheel spinning at a constant speed. As a seat moves around the wheel, its height above the ground goes up, then comes back down, then up again — forever repeating the same pattern. This repeating up-and-down movement is exactly what a sine or cosine graph looks like.

Functions that repeat the same pattern over and over are called periodic functions. Trigonometric functions are the most important examples of periodic functions.


2. The Basic Graphs: y=sinxy = \sin x, y=cosxy = \cos x, and y=tanxy = \tan x

Before looking at more complex functions, you need to know the basic (standard) shapes of these three graphs.


🔵 Graph of y=sinxy = \sin x

  • Starts at the origin (0°,0)(0°, 0)
  • Rises to a maximum of 1 at x=90°x = 90°
  • Returns to 0 at x=180°x = 180°
  • Falls to a minimum of −1 at x=270°x = 270°
  • Returns to 0 at x=360°x = 360°
  • Then the whole pattern repeats

Key facts:

PropertyValue
Amplitude1
Period360° or 2π2\pi radians
Maximum value1
Minimum value−1

🔵 Graph of y=cosxy = \cos x

  • Starts at a maximum of 1 at x=0°x = 0°
  • Falls to 0 at x=90°x = 90°
  • Reaches a minimum of −1 at x=180°x = 180°
  • Returns to 0 at x=270°x = 270°
  • Back to 1 at x=360°x = 360°
  • Then the whole pattern repeats

Key facts:

PropertyValue
Amplitude1
Period360° or 2π2\pi radians
Maximum value1
Minimum value−1

💡 Notice: The cosine graph looks exactly like the sine graph but shifted 90° to the left. They have the same shape, just starting at a different point.

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