Trigonometric Functions

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Introduction to Trigonometric Graphs

You already know about the trigonometric ratios: sine, cosine, and tangent. You've used them to find missing sides and angles in right-angled triangles.

Now we're going to look at what happens when we treat the angle as a variable (like x) and plot graphs of these functions. The angle x can be any value from 0° up to 360° (a full rotation), not just acute angles.

These graphs have special patterns that repeat over and over. Understanding these patterns will help you solve equations and recognize the graphs in exams.

The Graph of y = sin x

What does the sine graph look like?

The graph of y = sin x creates a smooth, wave-like shape. Here are the key features you need to know:

Key Features:

• Shape: A smooth wave that goes up and down
• Starting point: The graph passes through the origin (0, 0)
• Maximum value: The highest point on the wave is 1
• Minimum value: The lowest point on the wave is -1
• The graph is periodic: This means the pattern repeats itself. For sine, the pattern repeats every 360°

Important values to remember:

At specific angles, the sine graph reaches important points:

• At 0°: sin 0° = 0
• At 90°: sin 90° = 1 (maximum)
• At 180°: sin 180° = 0
• At 270°: sin 270° = -1 (minimum)
• At 360°: sin 360° = 0

You can remember this sequence: 0, 1, 0, -1 (and then it starts again).

How to sketch y = sin x:

1. Draw your axes. Mark the y-axis from -1 to 1. Mark the x-axis with 0°, 90°, 180°, 270°, and 360°
2. Plot the key points: (0, 0), (90, 1), (180, 0), (270, -1), (360, 0)
3. Join these points with a smooth, flowing curve (not straight lines!)
4. The curve should look like a wave

Symmetry: The sine wave is symmetrical. If you fold it at x = 90° or x = 270°, the two halves would match up. This symmetry is very useful when solving equations.