It's the circle of radius 1 centered at the origin. For an angle measured from the x-axis, the point on the circle has coordinates (cos θ, sin θ), giving the trig functions a geometric meaning.
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Sine, cosine, tangent and inverses with the unit circle.
Trigonometric functions relate an angle to the coordinates of a point on the unit circle (radius 1): cosine is the x coordinate and sine is the y coordinate of the point on the circle. The other four derive from those two, tangent = sin ÷ cos, cotangent = cos ÷ sin, secant = 1 ÷ cos and cosecant = 1 ÷ sin, and are undefined where the denominator is zero (tangent, for instance, does not exist at 90° and 270°, where cosine is 0). This calculator accepts the angle in degrees or radians, normalizes it to the range [0°, 360°), identifies the quadrant, computes the reference angle (the acute angle to the x-axis) and draws the angle on the unit circle. In inverse mode it returns the angle from a value using arcsin, arccos or arctan, respecting the domain: arcsin and arccos only accept values between −1 and 1. All results use 64-bit floating point, and values extremely close to 0 or ±1 are snapped to the exact value. Everything runs locally in your browser.
Mark an angle θ from the positive x-axis and follow it to the circle of radius 1: the point where you land has coordinates (cos θ, sin θ). That is the geometric definition, cosine measures the horizontal displacement and sine the vertical one. Because the radius is 1, the fundamental identity sin²θ + cos²θ = 1 always holds.
The other functions are combinations of those two. Tangent is the slope, tan θ = sin θ ÷ cos θ; cotangent is its reciprocal, cos θ ÷ sin θ; secant is 1 ÷ cos θ and cosecant is 1 ÷ sin θ. To convert units, use radians = degrees × π ÷ 180, so 180° equals π and 90° equals π/2.
Example 1, functions of 30°: in radians, 30 × π ÷ 180 ≈ 0.5236. Then sin 30° = 0.5, cos 30° ≈ 0.8660 and tan 30° = 0.5 ÷ 0.8660 ≈ 0.5774. The point on the circle is (0.8660, 0.5), in quadrant 1, and the reference angle is 30° itself.
Example 2, inverse function arccos(0.5): since 0.5 is within the domain [−1, 1], the tool looks for the angle whose cosine is 0.5 and returns arccos(0.5) = 60° (which is π/3 ≈ 1.0472 rad). Note that arccos always returns an angle between 0° and 180°.
Some functions do not exist at certain angles: tangent and secant blow up at 90° and 270° (cosine zero), while cotangent and cosecant blow up at 0° and 180° (sine zero), the tool flags these as undefined. The inverses return only the principal value: arcsin lies in [−90°, 90°], arccos in [0°, 180°] and arctan in (−90°, 90°); since sine and cosine repeat, infinitely many angles share the same value, and you recover the rest by adding the periods.
Trigonometry is the language of everything that oscillates or rotates: sound waves and electrical signals, circular motion, force analysis on ramps and structures, navigation and surveying (triangulation), computer graphics and animation. The reference angle, together with the quadrant sign rule (All, Sine, Tangent, Cosine), lets you obtain any value from the first quadrant.
Paste the code into your HTML and the tool shows up on your page, without J-Kit's navigation and ads. It still runs in the browser of whoever visits your site.
<iframe
src="https://jkit.tools/embed/en-US/trigonometry-calculator"
width="100%"
height="600"
style="border:0"
loading="lazy"
title="Trigonometry Calculator"
></iframe>These references help contextualize formulas, standards, APIs and limitations used on this page. They do not replace professional validation when a result has legal, financial, medical or operational impact.
It's the circle of radius 1 centered at the origin. For an angle measured from the x-axis, the point on the circle has coordinates (cos θ, sin θ), giving the trig functions a geometric meaning.
Calculations run in your browser. No data is sent to a server.