Tan Graph Made Simple: Learn How to Graph tan x

The tan graph is one of the three main trigonometric graphs, alongside sine and cosine. Understanding how to plot the graph of tan x helps students visualize the tangent function’s unique behavior, its periodicity, asymptotes, and symmetry.

In this tutorial, we explore how to graph y = tan x, recognize its key features, and compare it with sin, cos, tan graphs.

What Is the Tan Graph?

The tangent graph represents how the ratio of sine to cosine changes as the angle increases, and its shape reflects this relationship. Because cosine appears in the denominator, the graph behaves very differently from the smoother sin and cos curves, creating a repeating pattern of steep rises and falls. These characteristics give the tangent graph its signature look and make it an important function for studying angles, slopes, and periodic relationships.

As the angle approaches certain key values, the cosine of that angle approaches zero, causing the tangent value to grow extremely large in either the positive or negative direction. This is what produces the vertical asymptotes that break the graph into separate repeating segments. Understanding this behavior helps make sense of why the tangent graph looks more dramatic and less uniform than the other basic trigonometric graphs.

The tan graph represents the function y = tan x, where tangent is defined as sin x / cos x. Because cosine can equal zero at specific points, the graph of tan x has vertical asymptotes (undefined values) wherever cos x = 0, such as at 90°, 270°, etc.

Key characteristics of the graph of tangent function:

• It repeats every 180° (π radians).
• It passes through the origin (0, 0).
• The graph increases from negative to positive infinity between asymptotes.
• The range of tan x is all real numbers.
• The domain excludes values where cos x = 0.

How to Graph Tan x Step-by-Step

Before plotting points or adding asymptotes, it helps to understand the overall structure of the tangent curve. Each interval between asymptotes forms one repeating pattern called a branch, and every branch has the same general shape. By identifying one interval, marking its midpoint, and observing how the ratio sin ÷ cos changes across that interval, you build a reliable framework for the entire graph.

Follow these steps to plot y = tan x accurately:

1. Set up your axes: Label the x-axis with angles in degrees or radians (e.g., -180°, -90°, 0°, 90°, 180°).
2. Plot asymptotes: Draw dashed vertical lines at x = ±90°, ±270°, etc., where tan x is undefined.
3. Mark key points:
   • tan 0° = 0
   • tan 45° = 1
   • tan 135° = -1
4. Sketch the curves: Between each pair of asymptotes, draw an S-shaped curve that rises steeply through the origin or midpoint.
5. Repeat for periodicity: The pattern repeats every 180° (π radians).

Angle (°)	tan x
-135	1
-90	Undefined
-45	-1
0	0
45	1
90	Undefined
135	-1
180	0

This process helps you understand how graphing tan functions visually shows the relationship between sine and cosine.

Plot the above graph in Desmos by entering the expression: y = tan x

Click the wrench or spanner icon near the top right of the Desmos window. Set the X-Axis range to -450 to 450 and the Step to 45. Set the Y-Axis Step to 1. The following image shows the settings in Desmos.

Comparing Sin, Cos, and Tan Graphs

Although sine, cosine, and tangent are all derived from the unit circle, their graphs tell very different visual stories. Sine and cosine produce smooth waves with consistent peaks and troughs, while tangent shows much sharper behavior with values rising and falling rapidly. Comparing them side by side helps highlight these distinctions and reveals the mathematical relationships that connect the three functions.

The sine, cosine, and tangent functions are the three fundamental trigonometric functions, each with unique properties that show up clearly in their graphs. Plotting all three together helps students visualize the relationships between these functions, as the following image shows.

Key Differences Between Graphs

Before looking at each specific difference, it’s helpful to consider why these graphs behave differently at all. Each function takes information from the unit circle in a unique way, sine measures vertical movement, cosine measures horizontal movement, and tangent compares the two. This difference in origin is what creates the contrast in shape, period, and range.

1. Sine Graph (y = sin x)

• Shape: Smooth, wavelike curve oscillating between –1 and 1.
• Period: 360° (2π radians)
• Crosses the x-axis at multiples of 180° (0°, 180°, 360°…)

2. Cosine Graph (y = cos x)

• Shape: Similar to sine but shifted 90° to the left (or π/2 radians).
• Period: 360° (2π radians)
• Crosses the x-axis at odd multiples of 90° (90°, 270°, …)

3. Tangent Graph (y = tan x)

• Shape: S-shaped curve that rises from negative to positive infinity between asymptotes.
• Period: 180° (π radians)
• Undefined where cos x = 0, creating vertical asymptotes at 90°, 270°, etc.

How the Graphs Relate

Even though the graphs look different, they come from the same circular motion. As a point moves around the unit circle, the sine and cosine values change smoothly, and tangent reflects how steeply sine is rising or falling compared to cosine at each position. This connection makes tangent a natural extension of the behavior seen in the other two graphs.

The tangent function can be thought of as the ratio of sine to cosine:

tan(x) = sin(x) / cos(x)

This relationship explains why the tan graph has asymptotes wherever the cosine graph crosses zero.

While sine and cosine oscillate smoothly between –1 and 1, tangent increases without bound between asymptotes, highlighting its unique behavior.

Tips for Graphing Tan Graph

A successful tangent graph starts with identifying its asymptotes, since they define the structure of the curve. Once those boundaries are set, the middle of each interval becomes a guide point where the graph always crosses the x-axis. From there, the curve smoothly increases on one side and decreases on the other, giving the graph its characteristic “S-shaped” branches. Taking your time to sketch these guide points helps keep the graph accurate and consistent.

• Always identify where cos x = 0, since those points create asymptotes.
• Sketch one full period first, then extend it both directions.
• Use a graphing tool like Desmos to confirm your sketch.
• When comparing sin, cos, tan graphs, note that tangent has a shorter period and unbounded range.

Common Mistakes When Graphing Tan Graph

Many students struggle with tangent graphs because they try to draw them like sine or cosine curves. Forgetting the vertical asymptotes or placing them in the wrong positions is one of the most frequent errors. Another common issue is assuming the curve has a maximum or minimum value, when in fact tangent increases and decreases without bound. Recognizing these pitfalls early can make graphing much clearer.

1. Forgetting asymptotes – Beginners often skip vertical lines at 90°, 270°, etc.
2. Incorrect period – Remember: the tangent function repeats every π radians (180°), not 360°.
3. Wrong shape direction – Each section of the graph y = tan x increases left to right; it never decreases.

Other Interpretations of Tan Graph

The tangent graph can also be viewed as a way of measuring slope. At any angle, tan x represents the slope of a line that forms that angle with the positive x-axis. This interpretation makes tangent especially useful in fields like physics, engineering, and geometry, where angles and rates of change play a central role.

In trigonometry, tangent is also connected to right-triangle ratios, where tan x equals the opposite side divided by the adjacent side. Seen this way, the graph becomes a visual map of how this ratio changes as the angle opens and closes.

When someone searches for a tan graph, they might also mean:

• A graph of tan x in degrees vs radians.
• A graph of tangent function transformations such as y = a tan (bx + c).
• A combined sin cos tan graphs comparison chart to study phase differences.

Did You Know?

The tangent graph has been used for centuries as a tool for navigation and astronomy. Early astronomers relied on tangent tables to calculate the height of celestial objects above the horizon long before calculators existed.

Another interesting fact is that the word “tangent” comes from the Latin word tangens, meaning “touching.” This refers to the geometric idea of a tangent line touching a circle at exactly one point. Although the tangent function isn’t directly about touching lines, the historical name has remained.

The tangent function was originally defined using right triangles in ancient trigonometry. However, its graph form became more widely studied in the 17th century with the development of the unit circle. The tan graph has symmetry about the origin, making it an odd function, meaning tan(–x) = –tan(x).

Frequently Asked Questions About Tan Graphs

What is the period of the tan graph?

The period of tan x is 180° (π radians). The tangent graph repeats every 180° (π radians). This shorter period contrasts with sine and cosine, which repeat every 360° (2π radians). Understanding this difference helps when graphing multiple cycles.

Why does the tan graph have asymptotes?

Because tan x = sin x / cos x, and tangent is undefined when cos x = 0, creating vertical asymptotes. Asymptotes appear wherever cosine equals zero, because dividing by zero is undefined. At angles such as 90°, 270°, and their equivalents in radians (π/2, 3π/2, etc.), the tangent value increases or decreases without bound, which produces the vertical asymptotes seen on the graph.

How does the tan graph differ from sin and cos graphs?

Unlike sine and cosine, the tan graph is unbounded and repeats every 180°, not 360°. Sine and cosine are smooth curves with fixed maximum and minimum values, while tangent has no upper or lower limit and includes vertical asymptotes. This gives the tangent graph a more dramatic shape, with rapid increases and decreases between each pair of asymptotes.

Can the tan graph have maximum or minimum values?

No. The tangent function approaches infinity, so it doesn’t have fixed maximum or minimum points.

What happens when you graph y = tan 2x or y = tan (x – 45°)?

These are transformations of the graph of tan x, the first compresses horizontally, and the second shifts the graph horizontally.

Why does the tangent graph rise on one side and fall on the other?

This behavior happens because the sine value increases at a different rate than cosine near those angles. As cosine moves toward zero, the ratio sin ÷ cos becomes very large in the positive or negative direction, creating the characteristic rising and falling pattern.

Related Graph Types to Explore

When learning about tangent graphs, it helps to understand the broader family of graphs that commonly appear in trigonometry and algebra. Exploring related graph types allows you to see how functions behave differently, how they connect to one another, and how their shapes reflect underlying mathematical ideas. This broader perspective can make the characteristics of the tangent graph easier to recognize and compare, especially when studying transformations, periodicity, and real-world applications.

Other graphs, such as sine, cosine, reciprocal trig functions, or even non-trigonometric curves like exponential or logarithmic graphs, can offer useful reference points. By studying these alongside the tangent graph, you build stronger graph-reading skills and gain a deeper understanding of how each function behaves under changes in angle, amplitude, or domain.

If you’re learning trigonometric graphs, check out these related tutorials:

• Sine Graph (y = sin x)
• Cosine Graph (y = cos x)
• Secant and Cosecant Graphs

These help visualize relationships between the three main sin, cos, tan graphs.

Conclusion

The tan graph shows the repeating and unbounded nature of the tangent function, helping students understand periodicity and asymptotes. Whether you’re sketching the graph of tan x by hand or using a graphing calculator, mastering this function provides a foundation for advanced trigonometry. Use these tips and examples to make graphing tan functions simpler and more intuitive.

To explore more trigonometric functions and see how the sine, cosine, and tangent curves compare, visit our Trigonometry Graphs category for detailed tutorials and examples.