What is the unit circle?

The unit circle is a circle of radius 1 centred at the origin of the coordinate plane. For any angle θ measured counter-clockwise from the positive x-axis, the point where the radius meets the circle has coordinates (cos θ, sin θ). That single geometric fact defines all six trigonometric functions: sin = y, cos = x, tan = y/x, csc = 1/y, sec = 1/x, cot = x/y.

How does the calculator handle angles outside 0–360°?

All input angles are normalised to the range [0°, 360°) before calculation. For example, 450° becomes 90°, and −30° becomes 330°. Trig functions are periodic, so sin(450°) = sin(90°) = 1. The diagram always shows the equivalent angle in the first full revolution.

What is a reference angle?

The reference angle is the acute angle (0°–90°) between the terminal side of θ and the nearest part of the x-axis. In quadrant I it equals θ; in quadrant II it is 180° − θ; in quadrant III it is θ − 180°; in quadrant IV it is 360° − θ. Reference angles are useful because the magnitude of a trig function equals the value of that function at the reference angle — only the sign changes by quadrant.

Why is tan undefined at 90° and 270°?

Tan θ = sin θ / cos θ. At 90° cos θ = 0, so the division is undefined (the vertical asymptote of the tangent function). Similarly at 270°. The calculator shows "undefined" for tan, csc, sec, and cot wherever the denominator is zero, matching standard mathematical convention.

What is the Pythagorean identity and why does the calculator show it?

The Pythagorean identity states sin²θ + cos²θ = 1 for every angle θ, which follows directly from the Pythagorean theorem applied to the unit circle (the hypotenuse is always the radius = 1). The calculator computes and displays this sum as a floating-point sanity check — it will read 1.0000000000 (or extremely close due to IEEE 754 rounding) for any valid angle.

How do I use this to solve a triangle?

If you know one angle θ and the hypotenuse H of a right triangle, the opposite side is H·sin θ and the adjacent side is H·cos θ. If you know the opposite side O and the hypotenuse, enter sin(θ) = O/H in the "Enter sin value" mode to find θ. From θ you can derive all remaining sides and angles of the triangle.

Unit Circle Calculator

The unit circle is the single most important diagram in trigonometry. It is a circle of radius 1 centred at the origin. For any angle θ, the terminal point on the circle has x-coordinate cos θ and y-coordinate sin θ — everything else in trigonometry follows from that one fact. This calculator makes the relationship interactive: type any angle and every trig value, related angle, quadrant, and diagram updates in real time, entirely in your browser.

How it works

Place a point at angle θ measured counter-clockwise from the positive x-axis. Drop a vertical line from that point to the x-axis (length = |sin θ|) and draw the horizontal leg from the origin to the foot of that line (length = |cos θ|). You have a right triangle with hypotenuse 1 (the radius). The Pythagorean theorem immediately gives:

sin²θ + cos²θ = 1

The other four functions are reciprocals or ratios of sin and cos:

Function	Definition	Undefined when
tan θ	sin θ / cos θ	cos θ = 0 (90°, 270°)
csc θ	1 / sin θ	sin θ = 0 (0°, 180°, 360°)
sec θ	1 / cos θ	cos θ = 0 (90°, 270°)
cot θ	cos θ / sin θ	sin θ = 0 (0°, 180°, 360°)

The calculator normalises every input to [0°, 360°) — so 450° and 90° give identical results. To find an angle from a known ratio, switch to “Enter sin value” or “Enter cos value” mode; the calculator runs the inverse function (arcsin or arccos) and returns the principal value in [0°, 360°).

Worked example

Problem: A ramp rises at 35° to the horizontal. What fraction of the ramp length is the vertical rise?

Using the unit circle, the vertical component of a unit-length ramp at 35° is sin 35°.

Enter 35 in degrees mode.
Read: sin 35° ≈ 0.573576 — so the vertical rise is about 57.4% of the ramp length.
The horizontal run is cos 35° ≈ 0.819152 — about 81.9% of the ramp length.
Sanity check: 0.573576² + 0.819152² ≈ 1 (Pythagorean identity confirms the calculation).

Notable exact values (the ones worth memorising):

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	undef

Click any row in the notable-angles table inside the calculator to load it instantly.

Formula note

All six functions are defined from first principles at every angle the calculator accepts:

sin θ and cos θ are read directly from the (x, y) coordinates on the unit circle.
tan θ = sin θ / cos θ — geometrically, the length of the tangent segment from the point (1, 0) to where the extended radius meets the vertical tangent line at x = 1.
csc, sec, cot are their respective reciprocals.
Angles outside [0°, 360°) are reduced modulo 360° (modulo 2π for radians), exploiting the periodicity of all six functions.
Inverse modes use JavaScript’s built-in Math.asin and Math.acos, which return the principal value in [−π/2, π/2] and [0, π] respectively, then normalise to [0°, 360°).

The SVG diagram draws the unit circle at radius 90 px, the blue dashed horizontal leg (= |cos θ|), the red dashed vertical leg (= |sin θ|), and the purple radius arm, updating live as you type. The point coordinates shown on the diagram are the actual cos/sin values rounded to three decimal places.