Unit Circle With Tangent – Definition With Examples
Updated on August 4, 2026
On the unit circle, the tangent of an angle equals the slope of the line forming the terminal side of the angle when the angle is drawn from the origin.
What Is Tan on the Unit Circle?
The unit circle has a single unit as its radius. The tangent is one of the complex of three basic trigonometric functions (sine, cosine, and tangent) that help us understand and solve problems relating to an angle.
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Definition of the Unit Circle
The unit circle has a radius of one unit. This means that if we draw a line from the center of the circle to any point on the circle, that line has a length of one. Because the radius is fixed at 1, every point on the unit circle is exactly one unit away from the center.
The unit circle is usually centered at the origin (0, 0) on the coordinate plane, and we can use it to define sine and cosine using coordinates.
Definition of Tangent
In geometry, a tangent is a line that touches a circle at exactly one point. In trigonometry, however, tangent is a function defined as the ratio of sine to cosine.
How to Find Tan on the Unit Circle
To find tan on the unit circle, remember that it is not a separate coordinate but a ratio of two familiar ones. On the unit circle, cos(θ) is the x-coordinate of a point and sin(θ) is the y-coordinate. Since tangent is the ratio of sine to cosine, tan(θ) = sin(θ) / cos(θ).
In other words, if the point on the unit circle for angle θ is (x, y), then tan(θ) = y/x. We use the tangent to measure the steepness of an angle. When the x-coordinate (cos θ) equals zero, we cannot calculate this ratio, which is why tangent is undefined at 90° and 270°.
Properties of the Tan Unit Circle
Properties of the Unit Circle
- The circle has a single unit for the radius.
- Every point on the circle corresponds to coordinates (cos θ, sin θ).
- The x-coordinate is the cosine of the angle, and the y-coordinate is the sine of the angle.
- One full rotation equals 360° or 2π radians.
Properties of Tangent
- The tangent is a line that touches the circle at a single point.
- At the point of contact of the line and the circle, a right angle is formed.
Relationship Between the Unit Circle and Tangent
We can express the standard angles from 0° to 360° of the unit circle, including the tangent. This differs from the typical use of sine and cosine. We arrive at the tangent value by dividing the sine by the cosine — that is, the y-coordinate by the x-coordinate.
Chart of the Unit Circle With Tangent
You can make the conversion easier using the tangent chart below. It contains the values of sine and cosine converted from degrees and radians, so you don’t have to work them out manually every time.
Next, we convert sine and cosine to tangent by dividing one by the other. Here is the tangent chart with the unit circle values:
In tabular form, the unit circle with tangent looks like this:
The same values often appear arranged inside a circular chart. If you come across one that looks like the image below, don’t be surprised — it’s the same information:
Reading the chart, when the angle is 120° (2π/3 radians) the tangent is −√3; when the angle is 90° (π/2), the tangent is undefined, and so on.
The chart also shows that the unit circle divides into four quadrants:
- The first quadrant holds 0° to 90° (0 to π/2), where sine, cosine, and tangent are all positive.
- The second quadrant holds 90° to 180° (π/2 to π), where sine is positive and cosine is negative, making tangent negative.
- The third quadrant holds 180° to 270° (π to 3π/2).
- The fourth quadrant holds 270° to 360° (3π/2 to 2π).
Tangent is undefined at 90° (π/2) and 270° (3π/2) because cosine is 0 at both points. Since dividing by zero is not possible, the tangent does not exist there — the circle’s coordinates are (0, 1) at 90° and (0, −1) at 270°.
Equations of the Unit Circle and Tangent
Writing Equations of the Unit Circle
Derived from the Pythagorean theorem, the unit circle equation x² + y² = 1 is used to find the points (the x- and y-coordinates) that are one unit away from the center of the circle.
Writing Equations of Tangent
The tangent equation is tan(θ) = sin(θ) / cos(θ). Remember that on the unit circle sine is the y-coordinate and cosine is the x-coordinate, because each point of the circle is expressed as (x, y).
How to Remember the Unit Circle With Tangents
There are a few shortcuts for remembering tangent values without memorizing the whole chart. You only need to master the first quadrant and apply tan(θ) = sin(θ)/cos(θ), which is the ratio of the y-coordinate to the x-coordinate.
In the first quadrant the values follow a logical small, middle, large progression: at 30° (π/6) the tangent is √3/3, at 45° (π/4) it is exactly 1, and at 60° (π/3) it is √3.
For the rest of the circle these values repeat their numerical sequence, but you apply the ASTC (All-Sine-Tangent-Cosine) rule to determine the sign, since tangent is positive only in Quadrants I and III. Finally, tan(0°) = 0 and tan(180°) = 0 because the slope is flat on the horizontal axis, while tan(90°) and tan(270°) are undefined because the run (cos θ) is zero on the vertical axis.
Practice Problems on the Unit Circle and Tangent
Solved problem 1. Which statement correctly describes where a 45° angle with a tangent value of 1 is located? Answer: it is located in the first quadrant, where sine, cosine, and tangent are all positive.
Unit Circle With Tangent Worksheets
Put this into practice with related Brighterly worksheets:
- Unit circle printable
- Trigonometry worksheets
- Special right triangles worksheets
- 45-45-90 and 30-60-90 triangles worksheets
Unit Circle With Tangent: Practice Math Problems
Frequently Asked Questions on the Unit Circle With Tangent
What Is the Tangent on the Unit Circle?
On the unit circle, the tangent of an angle θ is the ratio of sine to cosine: tan(θ) = sin(θ) / cos(θ). Because each point on the circle has coordinates (cos θ, sin θ), this is the same as dividing the y-coordinate by the x-coordinate. Tangent also represents the slope of the angle’s terminal side drawn from the origin.
Why Is Tangent Undefined at 90° and 270°?
Tangent is undefined at 90° and 270° because the cosine (the x-coordinate) is 0 at those points — the coordinates are (0, 1) and (0, −1). Since tangent equals sine divided by cosine, computing it there would mean dividing by zero, which has no defined value. On a graph, the tangent function has vertical asymptotes at these angles.
What Is Tan(45°) on the Unit Circle?
The tangent of 45° is exactly 1. At 45° the point on the unit circle is (√2/2, √2/2), so tan(45°) = sin(45°) / cos(45°) = (√2/2) ÷ (√2/2) = 1. This is the middle value in the first quadrant, between tan(30°) = √3/3 and tan(60°) = √3.
In Which Quadrants Is Tangent Positive?
Tangent is positive in Quadrants I and III and negative in Quadrants II and IV. This is because tangent is the ratio of sine to cosine: it is positive only when sine and cosine share the same sign, which happens in the first quadrant (both positive) and the third quadrant (both negative). The ASTC rule captures this pattern.
What Is the Equation of the Unit Circle?
The equation of the unit circle is x² + y² = 1. It comes from the Pythagorean theorem applied to a radius of length 1, and it describes every point exactly one unit from the center at the origin. Any point (x, y) on the circle can also be written as (cos θ, sin θ) for its angle θ.
How Is Tangent Different From Sine and Cosine?
Sine and cosine are read directly from a point’s coordinates on the unit circle — sine is the y-coordinate and cosine is the x-coordinate. Tangent is not a coordinate; it is the ratio of the two, tan(θ) = sin(θ) / cos(θ) = y / x. That is why tangent can be undefined (when cosine is 0) while sine and cosine always stay between −1 and 1.
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