Unit Circle Chart- Tan and Cotangent Values
What the Unit Circle Actually Is
The unit circle is just a circle with a radius of exactly 1, centered at the origin of a coordinate plane. That's it. No tricks, no hidden complexity. Every point on this circle can be described using coordinates (cos θ, sin θ), where θ is the angle measured from the positive x-axis.
Most students memorize sin and cos values because teachers focus on them. But tan and cotangent are just as important—and far less understood. This guide fixes that.
Understanding Tangent (Tan) on the Unit Circle
Tangent = sin θ / cos θ
This ratio gives you the slope of the line from the origin to any point on the unit circle. When cos θ = 0, tangent is undefined. This happens at π/2 and 3π/2—90° and 270°.
Tan is positive when both sin and cos share the same sign. Tan is negative when they have opposite signs.
Where Tan Equals Key Values
- Tan = 0 when sin = 0 (angles: 0, π, 2π)
- Tan = 1 when sin = cos (angles: π/4, 5π/4)
- Tan = -1 when sin = -cos (angles: 3π/4, 7π/4)
- Tan is undefined at π/2 and 3π/2
Understanding Cotangent (Cot) on the Unit Circle
Cotangent = cos θ / sin θ = 1 / tan θ
Cot is the reciprocal of tan. When tan = 0, cot is undefined. This happens at 0, π, and 2π.
The signs of cot follow the same pattern as tan—positive when sin and cos agree, negative when they don't.
Where Cot Equals Key Values
- Cot = undefined when sin = 0 (angles: 0, π, 2π)
- Cot = 1 when cos = sin (angles: π/4, 5π/4)
- Cot = -1 when cos = -sin (angles: 3π/4, 7π/4)
- Cot = 0 at π/2 and 3π/2
Complete Unit Circle Chart: Tan and Cotangent Values
| Angle (Radians) | Angle (Degrees) | Tan θ | Cot θ |
|---|---|---|---|
| 0 | 0° | 0 | Undefined |
| π/6 | 30° | 1/√3 ≈ 0.577 | √3 ≈ 1.732 |
| π/4 | 45° | 1 | 1 |
| π/3 | 60° | √3 ≈ 1.732 | 1/√3 ≈ 0.577 |
| π/2 | 90° | Undefined | 0 |
| 2π/3 | 120° | -√3 ≈ -1.732 | -1/√3 ≈ -0.577 |
| 3π/4 | 135° | -1 | -1 |
| 5π/6 | 150° | -1/√3 ≈ -0.577 | -√3 ≈ -1.732 |
| π | 180° | 0 | Undefined |
| 7π/6 | 210° | 1/√3 ≈ 0.577 | √3 ≈ 1.732 |
| 5π/4 | 225° | 1 | 1 |
| 4π/3 | 240° | √3 ≈ 1.732 | 1/√3 ≈ 0.577 |
| 3π/2 | 270° | Undefined | 0 |
| 5π/3 | 300° | -√3 ≈ -1.732 | -1/√3 ≈ -0.577 |
| 7π/4 | 315° | -1 | -1 |
| 11π/6 | 330° | -1/√3 ≈ -0.577 | -√3 ≈ -1.732 |
| 2π | 360° | 0 | Undefined |
How to Use This Chart
You don't need to memorize everything. Here's what actually matters:
- Memorize the four key angles first: 0°, 45°, 90°, 180°. These are your anchors.
- Know the patterns: Tan and cot values repeat every π (180°). Once you know 0° to 180°, you know the whole circle.
- Remember where things break: Tan explodes at 90° and 270°. Cot explodes at 0°, 180°, and 360°.
Tan vs Cot: Quick Comparison
| Property | Tan θ | Cot θ |
|---|---|---|
| Formula | sin θ / cos θ | cos θ / sin θ |
| Reciprocal | Cot θ | Tan θ |
| Zeroes | θ = 0, π, 2π | θ = π/2, 3π/2 |
| Undefined | θ = π/2, 3π/2 | θ = 0, π, 2π |
| Period | π | π |
Getting Started: Reading Tan Values from Coordinates
Here's the practical method to find tan at any angle:
- Find your angle on the unit circle
- Read the (cos, sin) coordinates at that point
- Divide sin by cos
- That's your tan value
Example: At 60° (π/3), the coordinates are (1/2, √3/2). Tan = (√3/2) ÷ (1/2) = √3 ≈ 1.732.
For cot, just flip it: divide cos by sin instead.
Common Mistakes to Avoid
- Confusing undefined with zero. Tan at 90° is undefined (infinite slope), not zero. Students mix these up constantly.
- Forgetting the sign. Tan at 135° is -1, not 1. Check which quadrant you're in.
- Using degrees in radian formulas. Pick one system and stick with it throughout your calculation.
When You'll Actually Need This
If you're in trigonometry or calculus, this comes up constantly. Derivatives of tan involve sec². Integrals involving tan need you to know the behavior around asymptotes. The unit circle isn't abstract—it shows you exactly what these functions do.
Keep this chart handy. You'll reference it more than you think.