Unit Circle Calculator
The unit circle calculator gives any angle's point (cos θ, sin θ), its quadrant and reference angle, and all six trig values, in degrees or radians and exactly where they are exact.
Point (cos θ, sin θ)
- Angle in degrees Radians x 180/π.
- 135°
- Angle in radians A multiple of π, kept exact. The decimal is rounded.
- 3π/4 ≈ 2.35619449
- Coterminal angle in [0°, 360°)
- 135° (3π/4)
- Quadrant Quadrant II: x is negative and y positive, so only sine and cosecant are positive.
- II
- Reference angle
- 45° (π/4)
- Point (cos θ, sin θ) Exact. As decimals: (-0.7071067812, 0.7071067812).
- (-√2/2, √2/2)
- sin θ
- √2/2 ≈ 0.7071067812
- cos θ
- -√2/2 ≈ -0.7071067812
- tan θ
- -1
- csc θ
- √2 ≈ 1.414213562
- sec θ
- -√2 ≈ -1.414213562
- cot θ
- -1
| Degrees | Radians | cos θ | sin θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 |
| 30° | π/6 | √3/2 | 1/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | 1/2 | √3/2 | √3 |
| 90° | π/2 | 0 | 1 | Undefined |
| 120° | 2π/3 | -1/2 | √3/2 | -√3 |
| 135° | 3π/4 | -√2/2 | √2/2 | -1 |
| 150° | 5π/6 | -√3/2 | 1/2 | -√3/3 |
| 180° | π | -1 | 0 | 0 |
| 210° | 7π/6 | -√3/2 | -1/2 | √3/3 |
| 225° | 5π/4 | -√2/2 | -√2/2 | 1 |
| 240° | 4π/3 | -1/2 | -√3/2 | √3 |
| 270° | 3π/2 | 0 | -1 | Undefined |
| 300° | 5π/3 | 1/2 | -√3/2 | -√3 |
| 315° | 7π/4 | √2/2 | -√2/2 | -1 |
| 330° | 11π/6 | √3/2 | -1/2 | -√3/3 |
How the Unit Circle Calculator works
The unit circle is the circle of radius 1 centered at the origin. Turn an angle θ counterclockwise from the positive x-axis and it meets the circle at one point, and that point is (cos θ, sin θ). That single fact is where trigonometry and the unit circle meet: cosine is the x-coordinate, sine is the y-coordinate, and the other four functions are ratios of the two. Tangent is y / x, cotangent is x / y, secant is 1 / x and cosecant is 1 / y.
A radian unit circle measures the angle by the arc it cuts off. On a circle of radius 1, an angle of 1 radian cuts off an arc of length 1, so a full turn is 2π radians and a half turn is π. That gives the conversion: 180° = π radians, so 135° is 135 x π/180 = 3π/4.
The 16 angles on every trig unit circle chart are the multiples of 30° and 45°: 0, π/6, π/4, π/3, π/2, and their mirror images in the other three quadrants. Their coordinates come from two triangles. A 45-45-90 triangle with a hypotenuse of 1 has legs of √2/2, so 45° is at (√2/2, √2/2). A 30-60-90 triangle with a hypotenuse of 1 has legs of 1/2 and √3/2, so 30° is at (√3/2, 1/2) and 60° is at (1/2, √3/2). Every other standard angle reuses these numbers with different signs.
The signs come from the quadrant. In quadrant I, x and y are positive, so every function is positive. In quadrant II, x is negative, so only sine and cosecant stay positive. In quadrant III both are negative, which leaves tangent and cotangent positive. In quadrant IV only cosine and secant are positive. The reference angle is the acute angle between the ray and the x-axis: 180° - θ in quadrant II, θ - 180° in quadrant III, and 360° - θ in quadrant IV. Find the values for the reference angle, give them the quadrant's signs, and you have the values for θ.
Angles outside one turn land on the same points. Adding or subtracting 360° (2π) gives a coterminal angle with the same position, so 855° and -225° both end where 135° does.
Because every point on the circle satisfies x² + y² = 1, the coordinates give the Pythagorean identity cos²θ + sin²θ = 1 for every angle.
Unit Circle Calculator FAQ
What is the unit circle?
The circle of radius 1 centered at the origin. An angle θ measured counterclockwise from the positive x-axis meets it at the point (cos θ, sin θ), so the circle shows the cosine and sine of every angle at once.
How do you remember the unit circle?
Learn the first quadrant and reuse it. The coordinates at 30°, 45° and 60° are built from 1/2, √2/2 and √3/2. In the other quadrants the same numbers appear with signs set by the quadrant: x is negative on the left and y is negative below.
What is 3π/4 on the unit circle?
135°, in quadrant II. Its reference angle is π/4, so the point is (-√2/2, √2/2): cos 3π/4 = -√2/2, sin 3π/4 = √2/2 and tan 3π/4 = -1.
Why is tan 90° undefined?
Tangent is sin θ / cos θ. At 90° the point is (0, 1), so cos 90° = 0 and the division has no answer. The same happens at 270°. Cotangent is undefined at 0° and 180°, where sin θ = 0.
How do I convert degrees to radians?
Multiply by π/180. 60° is 60π/180 = π/3, and 270° is 3π/2. To go back, multiply radians by 180/π: π/6 is 30°.