How to Graph the Tangent Function (Degrees) - Quick Guide

To graph y = tan(θ) in degrees, start with the identity:

tan(θ) = sin(θ)/cos(θ)

That fraction tells you exactly where the graph crosses the x-axis and where it has vertical asymptotes.

Step 1: Plot the x-intercepts (where the numerator is 0)

Tangent equals 0 when:

sin(θ) = 0

In degrees, that happens at:

θ = 0°, 180°, 360°, 540°, ...

In general:

θ = k ⋅ 180° where k is an integer

So the tangent graph crosses the x-axis at every multiple of 180°

Step 2: Plot the vertical asymptotes (where the denominator is 0)

Tangent is undefined when:

cos(θ) = 0

In degrees, that happens at:

θ = 90°, 270°, 450°, 630°

In general:

θ = 90° + k ⋅ 180° where k is an integer

So draw vertical dashed lines at those angles.

Step 3: Sketch one “tangent branch” between two asymptotes

Start with the main interval:

-90° < θ < 90°

Key facts:

• The curve passes through (0°, 0).

• As θ → -90° (from the right), tan(θ) → -∞

• As θ → 90° (from the left), tan(θ) → +∞

• The curve is increasing across the whole interval.

So draw a smooth increasing curve that drops down near -90° and shoots up near 90°.

Step 4: Decide above or below the x-axis using quadrant signs

Because tan(θ) = sin(θ)/cos(θ), its sign depends on the quadrant:

• Quadrant I (0° to 90°): (+) over (+) = (+) (above x-axis)

• Quadrant II (90° to 180°): (+) over (-) = (-) (below x-axis)

• Quadrant III (180° to 270°): (-) over (-) = (+) (above x-axis)

• Quadrant IV (270° to 360°): (-) over (+) = (-) (below x-axis)

This is a quick way to check whether each branch should be mostly above or below.

Step 5: Repeat the pattern (period 180°)

Tangent repeats every 180°:

tan(θ + 180°) = tan(θ)

So once you draw the branch from -90° to 90°, you can repeat that same shape every 180°, always between consecutive asymptotes.

Step 6: Graph the function

Here is the final graph of the tangent function on -360° ≤ θ ≤ 360°:

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