Trigonometric Graphs & Equations

The big idea: Sine, cosine, and tangent create repeating wave patterns. By sketching these graphs and understanding their symmetry, you can find all solutions to trigonometric equations — not just the one your calculator gives you.

• sin x and cos x repeat every 360°; they oscillate between –1 and 1
• tan x repeats every 180°; it shoots to infinity and has vertical asymptotes
• To solve any trig equation: (1) find the first solution using inverse functions, (2) sketch the graph, (3) use symmetry to find all other solutions in your range
• The 180° − x and 360° − x rules come directly from the symmetry of the graphs

Shape: The sine graph is a smooth, continuous wave that oscillates (bounces up and down) between a height of 1 and −1.

Key properties:

• Period: 360° — the wave completes one full cycle every 360°
• y-intercept: The graph passes through the origin, so sin 0° = 0
• Amplitude: The maximum distance from the centre is 1 (the graph never goes above 1 or below −1)
• Symmetry: If you know sin x = k, then sin(180° − x) = k as well. This is the critical rule for finding second solutions.

Key coordinate points (memorise these):

Shape: The cosine graph looks almost identical to the sine graph, but it's shifted 90° to the left. Like sine, it oscillates between 1 and −1 with a period of 360°.

Key properties:

• Period: 360°
• y-intercept: The graph starts at 1, so cos 0° = 1
• Amplitude: Maximum distance from centre is 1
• Symmetry: cos x = k cos(360° − x) = k