Unit Circle Calculator
Enter an angle (degrees or radians) to get quadrant, reference angle, sin/cos/tan, and the unit-circle point (cos θ, sin θ). Or flip it: enter a trig value (like sin θ = √2/2) and get all solutions in [0°, 360°) or [0, 2π).
Quick reminder
On the unit circle: x = cos(θ) and y = sin(θ).
And tan(θ) = sin(θ)/cos(θ) (undefined when cos(θ)=0).
Background
The unit circle connects angles to coordinates. Once you know the quadrant and reference angle, you can quickly get signs and (for special angles) exact values like √2/2 and √3/2.
How to use this calculator
- Choose a Mode: Angle → values or Trig value → angles.
- Use Quick picks for common angles or exact trig values.
- Turn on Exact values to see radicals for standard angles.
- Use the solution set range to list all answers in [0°, 360°) or [0, 2π).
How it works
- Angle mode: compute sin(θ), cos(θ), tan(θ) and the point (cosθ, sinθ).
- Inverse mode: use arcsin, arccos, or arctan to get a principal angle, then generate all coterminal solutions in the selected range.
- Reference angle and signs come from the quadrant.
Formula & Equation Used
Coordinates: (x, y) = (cos(θ), sin(θ))
Tangent: tan(θ) = sin(θ)/cos(θ)
Conversion: π rad = 180°
Normalization: θ ≡ θ + 360°k (or θ ≡ θ + 2πk)
Example Problem & Step-by-Step Solution
Example 1 — Solve sin(θ) = √2/2 on [0°, 360°)
- Recognize √2/2 as a standard unit-circle value (reference angle 45°).
- sin(θ) is positive in Quadrants I and II.
- Solutions: θ = 45° and θ = 135°.
Example 2 — θ = −210° (Angle → values)
- Normalize to [0°, 360°): −210° + 360° = 150°.
- Quadrant: 150° is in Quadrant II → sin positive, cos negative, tan negative.
- Reference angle: 180° − 150° = 30°.
- Exact values: sin(150°)=1/2, cos(150°)=−√3/2, tan(150°)=−√3/3.
- Unit-circle point: (cosθ, sinθ) = (−√3/2, 1/2).
Example 3 — Solve cos(θ) = −1/2 on [0°, 360°)
- 1/2 corresponds to a reference angle of 60°.
- cos(θ) is negative in Quadrants II and III.
- Solutions: θ = 180° − 60° = 120° and θ = 180° + 60° = 240°.
Frequently Asked Questions
Q: What is a reference angle?
The reference angle is the acute angle (between 0° and 90°) between the terminal side of θ and the x-axis.
Q: Why are some tan values “undefined”?
Because tan(θ)=sin(θ)/cos(θ), and dividing by zero is undefined. This happens at 90° and 270° (and their coterminal angles).
Q: Why do you sometimes show exact values?
Exact values are available for the standard unit-circle angles (multiples of 30° and 45°). For other angles we show accurate decimals.
Q: Why does sin(θ)=0.5 usually have two answers in 0°–360°?
Because a horizontal line intersects the unit circle in two places. In [0°,360°), most non-extreme trig values occur in two quadrants (depending on the sign).
Q: How do you get all solutions in the selected range?
We find a principal angle using arcsin, arccos, or arctan, then generate the other angle(s) in the range using symmetry + quadrant sign rules (and coterminal angles within the range).
Q: When does a trig equation have only one solution?
Some values hit the unit circle at a single angle in the chosen interval (for example, sin(θ)=1 at 90°, cos(θ)=−1 at 180°). Others can have two, and tan can be undefined at certain angles.
