10.3 Sketching Trigonometric Graphs

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📚 Main Content

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🔵 The Graph of y = sin x

• The sine graph starts at (0, 0), rises to a peak of 1 at x = 90°, comes back to 0 at x = 180°, drops to a trough of −1 at x = 270°, and returns to 0 at x = 360°.
• The graph is smooth and wave-shaped, like a rolling hill.
• It repeats the same pattern every 360° (or 2π radians). This length of one full repeat is called the period.
• The graph sits between y = −1 and y = 1. The distance from the middle (the x-axis) to the peak is called the amplitude, which is 1 for this basic graph.

Key points to plot for y = sin x over 0° to 360°:

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🔵 The Graph of y = cos x

• The cosine graph starts at (0, 1), drops to 0 at x = 90°, reaches its lowest point of −1 at x = 180°, rises back to 0 at x = 270°, and returns to 1 at x = 360°.
• Its shape is the same wave as sine — just shifted to the left by 90°.
• Period = 360° (or 2π radians). Amplitude = 1.

Key points to plot for y = cos x over 0° to 360°:

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🔵 The Graph of y = tan x

• The tangent graph behaves very differently from sine and cosine.
• It does not have an amplitude — it stretches to positive and negative infinity.
• It has vertical asymptotes: these are invisible vertical lines that the graph gets closer and closer to but never touches or crosses. For y = tan x, asymptotes occur at x = 90°, x = 270°, x = −90°, etc. — in other words, every 90° from 90° onwards.
• Period = 180° (or π radians). The graph repeats every 180°.
• Each branch of the graph rises from negative infinity to positive infinity between two consecutive asymptotes.

Key points for y = tan x (no amplitude, but note the asymptotes):