Tangent Unit Circle Values- Complete Reference Chart

What Is Tangent on the Unit Circle?

The unit circle is a circle with radius 1 centered at the origin. Every point on it follows the equation x² + y² = 1. Tangent at any angle is simply y/x — the ratio of the y-coordinate to the x-coordinate.

That ratio gives you the slope of the line from the origin through that point. When x = 0, you're dividing by zero. That's why tangent is undefined at π/2 and 3π/2 — the line is vertical and has no defined slope.

Why You Need This Reference Chart

Most students memorize the sine and cosine table and then scramble to derive tangent during exams. That's backwards. Tangent values follow a pattern you can learn in 5 minutes if you know the right relationships.

This chart gives you every key tangent value from 0 to 2π, in both radians and degrees, with exact values and decimal approximations.

Complete Tangent Unit Circle Values Chart

Angle (Radians)	Angle (Degrees)	Reference Angle	Exact Tangent Value	Decimal Approximation	Sign
0	0°	0°	0	0.0000	0
π/6	30°	30°	1/√3	0.5774	+
π/4	45°	45°	1	1.0000	+
π/3	60°	60°	√3	1.7321	+
π/2	90°	—	undefined	±∞	undefined
2π/3	120°	60°	-√3	-1.7321	-
3π/4	135°	45°	-1	-1.0000	-
5π/6	150°	30°	-1/√3	-0.5774	-
π	180°	0°	0	0.0000	0
7π/6	210°	30°	1/√3	0.5774	+
5π/4	225°	45°	1	1.0000	+
4π/3	240°	60°	√3	1.7321	+
3π/2	270°	—	undefined	±∞	undefined
5π/3	300°	60°	-√3	-1.7321	-
7π/4	315°	45°	-1	-1.0000	-
11π/6	330°	30°	-1/√3	-0.5774	-
2π	360°	0°	0	0.0000	0

The Pattern: Memorize Three Values

You don't need to memorize all 17 rows. You need to memorize three base values:

tan(30°) = 1/√3 ≈ 0.577
tan(45°) = 1
tan(60°) = √3 ≈ 1.732

Everything else follows from quadrant rules. Tangent is positive in Quadrants I and III, negative in Quadrants II and IV.

How to Find Any Tangent Value

Step 1: Find your angle's reference angle (the acute angle to the x-axis).

Step 2: Identify which quadrant you're in.

Step 3: Apply the correct base value with the appropriate sign.

Example: What is tan(225°)?

Reference angle is 45°. 225° is in Quadrant III. Tangent is positive in Quadrant III. tan(45°) = 1. Therefore, tan(225°) = 1.

Common Mistakes to Avoid

Confusing sine and tangent signs. Sine is positive in Quadrants I and II. Tangent is positive in Quadrants I and III. Don't mix them up.
Forgetting the undefined points. At π/2 and 3π/2, tangent does not exist. Writing a number here is mathematically wrong.
Rationalizing unnecessarily. Both 1/√3 and √3/3 are correct. Pick one and stick with it.
Using calculators in the wrong mode. Make sure your calculator is in the right mode (degrees vs radians) or you'll get garbage answers.

How to Use This Chart for Homework and Tests

Write out the three base values on your scratch paper first. Then sketch a quick unit circle with + and - signs in each quadrant:

Quadrant I: sin +, cos +, tan +
Quadrant II: sin +, cos -, tan -
Quadrant III: sin -, cos -, tan +
Quadrant IV: sin -, cos +, tan -

Combine the reference angle value with the correct sign, and you have your answer in under 10 seconds.

When Tangent Equals 0 vs. When It's Undefined

These get confused constantly.

Tangent equals 0 when the angle lands on the x-axis — at 0, π, and 2π. The y-coordinate is 0, so y/x = 0.

Tangent is undefined when the angle lands on the y-axis — at π/2 and 3π/2. The x-coordinate is 0, and you cannot divide by zero.

Quick Reference: Key Tangent Values

Angle	Tangent	Memory Trick
0°	0	Start at origin
30°	1/√3	Small positive
45°	1	Equal sides, equal values
60°	√3	Largest base value
90°	undefined	Vertical wall

Practical Applications

You need tangent values for:

Solving trigonometric equations
Calculus problems involving derivatives of trig functions
Physics problems with angles of elevation and slopes
Engineering calculations involving angles and ratios
Computer graphics and rotation matrices

If you're in a STEM course, this chart will show up repeatedly. The good news: once you learn the three base values and the quadrant sign rules, you can generate any tangent value from memory.