Tangent Graph Overview

Overview

This lesson explains how to sketch and analyze the tangent (tan) graph, discusses its key properties, and highlights differences from sine and cosine graphs.

Plotting the Tangent Graph

• Steps of 45 degrees are used for the tan graph instead of the 90-degree steps for sine and cosine.
• Standard x-values to plot include 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, and 360° (as well as their negative counterparts).
• Corresponding y-values at key points: tan 0° = 0, tan 45° = 1, tan 135° = -1, tan 180° = 0, tan 225° = 1, tan 315° = -1, tan 360° = 0.
• At 90°, 270°, -90°, -270°, tan is undefined, marked by vertical dashed asymptotes.
• The pattern repeats for negative angles, mirroring positive values.

Connecting the Graph

• Plot points and connect them to approach the asymptotes without crossing them.
• The resulting graph appears as repeating "snakes" between asymptotes, rising and falling rapidly near them.

Properties of the Tan Graph

• Period: For tan, the period is 180°, unlike 360° for sine and cosine; the graph repeats every 180°.
• Amplitude: The tan graph has no amplitude since it extends infinitely up and down (no maximum or minimum values).
• Domain: x ranges from -360° to 360°, except where tan x is undefined: x ≠ -270°, -90°, 90°, 270°.
• Range: y can be any real number; y ∈ (−∞, ∞).

Key Terms & Definitions

• Tangent graph (tan x) — A trigonometric graph repeating every 180°, undefined at odd multiples of 90°.
• Asymptote — A dashed vertical line representing values where the graph is undefined.
• Period — The interval before the graph repeats, 180° for tan.
• Amplitude — The maximum distance from the resting position, not applicable to tan.
• Domain — The set of allowable x-values for the function.
• Range — The set of possible output (y) values; (−∞, ∞) for tan x.

Action Items / Next Steps

• Practice plotting the tan graph using steps of 45° and marking asymptotes at odd multiples of 90°.
• Memorize that tan’s period is 180° and that its amplitude is not applicable.
• Review domain and range restrictions for the tan graph.