Unit Circle with Tangent- Trigonometry Guide and Examples
What Is the Unit Circle?
The unit circle is a circle with a radius of exactly 1 unit, centered at the origin (0, 0) on a coordinate plane. That's it. Nothing fancy.
Every point on this circle can be written as (cos θ, sin θ) where θ is the angle measured from the positive x-axis. This relationship is what makes the unit circle so useful for trigonometry.
The Tangent Function on the Unit Circle
Tangent is defined as the ratio of sin θ to cos θ:
tan θ = sin θ / cos θ
On the unit circle, this means tangent is the y-coordinate divided by the x-coordinate at any point.
Here's the catch: tangent is undefined whenever cos θ = 0. This happens at π/2 and 3π/2 (90° and 270°). The function shoots off to infinity at these points. You'll see vertical asymptotes on any tangent graph.
When Tangent Is Positive vs. Negative
Tangent inherits its sign from both sine and cosine:
- Both sin and cos positive → tan is positive (Quadrant I)
- Sin positive, cos negative → tan is negative (Quadrant II)
- Both sin and cos negative → tan is positive (Quadrant III)
- Sin negative, cos positive → tan is negative (Quadrant IV)
Key Angles and Their Tangent Values
You need to memorize these values. They're tested constantly.
| Angle (Degrees) | Angle (Radians) | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 = √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | Undefined |
| 180° | π | 0 | -1 | 0 |
| 270° | 3π/2 | -1 | 0 | Undefined |
How to Find Tangent on the Unit Circle
Follow these steps:
- Locate the angle on the unit circle
- Find the coordinates of that point — that's (cos θ, sin θ)
- Divide sin θ by cos θ
That's the whole process. If you're given a point (x, y) on the unit circle, tan θ = y/x.
Examples
Example 1: Find tan(π/4)
At π/4 (45°), the coordinates are (√2/2, √2/2).
tan(π/4) = (√2/2) / (√2/2) = 1
Example 2: Find tan(5π/6)
5π/6 is in Quadrant II. The coordinates are (-√3/2, 1/2).
tan(5π/6) = (1/2) / (-√3/2) = -1/√3 = -√3/3
Example 3: Find tan(π)
At π (180°), the coordinates are (-1, 0).
tan(π) = 0 / (-1) = 0
Common Mistakes
Students mess this up in predictable ways:
- Forgetting the undefined cases — Always check if cos θ = 0 before dividing
- Getting the sign wrong — Quadrant matters. Don't just memorize absolute values
- Confusing reference angles — The reference angle gives you the magnitude, but you still need the correct sign for the quadrant
- Rationalizing incorrectly — 1/√3 and √3/3 are equivalent. Know how to convert
Getting Started: Practice Method
If you're learning this for the first time:
- Draw the unit circle from memory — all four quadrants, key angles marked
- Write the coordinates for 0°, 30°, 45°, 60°, 90° and their equivalents in other quadrants
- Calculate tangent for each point by dividing y by x
- Check your signs against the quadrant rules
- Drill until you can do this in under 2 minutes without hesitation
Why This Matters
The unit circle isn't just an abstract concept. It shows up in:
- Physics — analyzing waves, oscillations, and rotations
- Engineering — signal processing, control systems
- Computer graphics — rotations, transformations
- Navigation — angles, bearings, coordinates
Once you understand the unit circle with tangent, you understand the foundation for all circular trigonometry. Everything else builds from this.