How this calculator works
The unit circle is a circle of radius 1 centered at the origin of a coordinate plane. For any angle θ measured counterclockwise from the positive x-axis, the point where the angle's terminal side crosses the circle has coordinates (cos θ, sin θ) — which is the entire reason cosine and sine are defined the way they are in trigonometry.
This calculator takes an angle in degrees or radians and returns sin, cos and tan, the (x, y) coordinate on the circle, which quadrant the angle falls in, and its reference angle (the acute angle to the nearest x-axis). For the 16 angles that are multiples of 30° or 45° — the ones every trigonometry course memorizes — it also shows the exact value (like √3/2) rather than just a decimal approximation.
The core relationships
Point on the circle: (x, y) = (cos θ, sin θ)tan θ = sin θ ÷ cos θ, undefined wherever cos θ = 0 (at 90° and 270°)Pythagorean identity: sin²θ + cos²θ = 1, for every angle, because every point (cos θ, sin θ) is exactly 1 unit from the originThe 16 standard angles — 0° (0): sin 0, cos 1, tan 030° (π/6): sin 1/2, cos √3/2, tan √3/3 — 45° (π/4): sin √2/2, cos √2/2, tan 1 — 60° (π/3): sin √3/2, cos 1/2, tan √3 — 90° (π/2): sin 1, cos 0, tan undefined120° (2π/3): sin √3/2, cos -1/2, tan -√3 — 135° (3π/4): sin √2/2, cos -√2/2, tan -1 — 150° (5π/6): sin 1/2, cos -√3/2, tan -√3/3 — 180° (π): sin 0, cos -1, tan 0210° (7π/6): sin -1/2, cos -√3/2, tan √3/3 — 225° (5π/4): sin -√2/2, cos -√2/2, tan 1 — 240° (4π/3): sin -√3/2, cos -1/2, tan √3 — 270° (3π/2): sin -1, cos 0, tan undefined300° (5π/3): sin -√3/2, cos 1/2, tan -√3 — 315° (7π/4): sin -√2/2, cos √2/2, tan -1 — 330° (11π/6): sin -1/2, cos √3/2, tan -√3/3Reference angle is the acute angle (between 0° and 90°) formed between the terminal side of θ and the x-axis; it is what you look up in a 30-60-90 or 45-45-90 triangle before applying the correct sign for the quadrant.
Worked example: 150 degrees
- 150° is in the second quadrant (between 90° and 180°), so x is negative and y is positive.
- Its reference angle is 180° − 150° = 30°, the same acute angle as the 30-60-90 triangle.
- Sine keeps the same value as the reference angle in quadrant II: sin 150° = sin 30° = 1/2. Cosine flips sign: cos 150° = −cos 30° = −√3/2.
- The point on the unit circle is (−√3/2, 1/2) ≈ (−0.866, 0.5), and tan 150° = (1/2) ÷ (−√3/2) = −√3/3 ≈ −0.577.
Frequently asked questions
Why is tangent undefined at 90 degrees and 270 degrees?
Because tan θ = sin θ ÷ cos θ, and cos 90° = cos 270° = 0. Dividing by zero is undefined, so tangent has a vertical asymptote at every angle where the point on the unit circle sits exactly on the y-axis.
How do I find the exact value instead of a decimal?
For the 16 standard angles (multiples of 30° and 45° from 0° to 330°), sin and cos are always 0, ±1/2, ±√2/2 or ±√3/2, and this calculator shows that exact form directly. Any other angle only has a decimal approximation, since its sine and cosine are generally irrational numbers with no simple closed form.
What is a reference angle used for?
It lets you find any angle's sine, cosine and tangent from a single memorized 30-60-90 or 45-45-90 triangle: look up the reference angle's value, then apply the sign (+ or −) that matches the quadrant the original angle is in.
How is this different from converting between degrees and radians?
A degrees-to-radians converter only changes the unit an angle is measured in. This calculator takes that angle and evaluates the trigonometric functions at it, which is a separate step — you can enter the angle here in either unit and get the same trigonometric result.
Why does the unit circle have radius 1?
A radius of 1 makes the Pythagorean identity read directly as sin²θ + cos²θ = 1, since every point on the circle is exactly distance 1 from the center — that simplicity is why sine and cosine are defined using this specific circle rather than an arbitrary one.