Unit Circle Calculator
Common Angle Values
| Degrees | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | ∞ |
| 180° | π | 0 | -1 | 0 |
| 270° | 3π/2 | -1 | 0 | ∞ |
| 360° | 2π | 0 | 1 | 0 |
The Unit Circle
Quadrant Signs
Related Calculators
Understanding the Unit Circle
What is the Unit Circle?
The unit circle is a circle with a radius of 1, centered at the origin (0, 0) of a coordinate plane. It's one of the most important concepts in trigonometry because it provides a geometric way to understand all trigonometric functions.
Fundamental Definition
For any angle θ measured from the positive x-axis, the point where the terminal side intersects the unit circle has coordinates (cos θ, sin θ).
The Six Trigonometric Functions
Primary Functions
Reciprocal Functions
Special Angles
Certain angles have exact trigonometric values that can be expressed without decimals. These "special angles" are essential to memorize:
| Degrees | Radians | sin θ | cos θ | tan θ | (x, y) |
|---|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 | (1, 0) |
| 30° | π/6 | 1/2 | √3/2 | √3/3 | (√3/2, 1/2) |
| 45° | π/4 | √2/2 | √2/2 | 1 | (√2/2, √2/2) |
| 60° | π/3 | √3/2 | 1/2 | √3 | (1/2, √3/2) |
| 90° | π/2 | 1 | 0 | undef | (0, 1) |
Degrees vs. Radians
Degrees
A full circle is divided into 360 degrees. This system comes from ancient Babylonian mathematics (360 ≈ days in a year).
Radians
One radian is the angle where the arc length equals the radius. More natural for calculus and physics.
Conversion Formulas
Quadrants and Signs
The unit circle is divided into four quadrants. The signs of trig functions depend on which quadrant the angle is in:
Pythagorean Identities
These identities come directly from the Pythagorean theorem and the unit circle definition.
Reference Angles
A reference angle is the acute angle (0-90°) formed between the terminal side and the x-axis. Any trig function of an angle equals ± the same function of its reference angle, with the sign determined by the quadrant.
Finding Reference Angles
Memory Tip
For special angle values, notice the pattern in sine values from 0° to 90°: √0/2, √1/2, √2/2, √3/2, √4/2. This gives 0, 1/2, √2/2, √3/2, 1. Cosine values follow the same pattern in reverse.
Frequently Asked Questions
What is the unit circle and why is it important?
The unit circle is a circle with radius 1 centered at the origin. It's fundamental because any point on it has coordinates (cos theta, sin theta), giving a geometric definition of trigonometric functions. All trig identities and relationships can be derived from it.
How do you remember the special angles on the unit circle?
For sine values from 0 to 90 degrees, remember: square root of 0/2, square root of 1/2, square root of 2/2, square root of 3/2, square root of 4/2. This gives 0, 1/2, root 2 over 2, root 3 over 2, and 1. Cosine follows the same pattern in reverse.
What does ASTC (All Students Take Calculus) mean?
It's a mnemonic for which trig functions are positive in each quadrant. Quadrant 1: All positive. Quadrant 2: Sine positive. Quadrant 3: Tangent positive. Quadrant 4: Cosine positive. This helps determine signs when evaluating trig functions.
How do you convert between degrees and radians?
Multiply degrees by pi/180 to get radians. Multiply radians by 180/pi to get degrees. Key conversions: 180 degrees = pi radians, 90 degrees = pi/2 radians, 60 degrees = pi/3 radians, 45 degrees = pi/4 radians, 30 degrees = pi/6 radians.