Find the six trigonometric functions of any angle, or solve a right triangle from its sides and angles. Choose a tab to get started.
Right angle at C. Enter enough values to define the triangle: two sides, one side and one acute angle, or one side and the area. Leave the rest blank.
The trigonometry calculator does two things in one place. First, it finds the values of all six trigonometric functions — sine, cosine, tangent, cotangent, secant, and cosecant — for any angle you enter, in degrees or radians. Second, it acts as a complete right triangle solver: give it any two measurements of a right triangle (two sides, a side and an angle, or a side and the area) and it works out every remaining side, angle, and the area. Whether you are studying for a math test, working through a physics problem, or laying out a construction project, this tool covers the everyday trigonometry you need.
Trigonometry is the branch of mathematics that connects the angles of a triangle to the lengths of its sides. Those connections turn out to be astonishingly useful: they let us measure distances we cannot reach, describe waves and oscillations, navigate across oceans, and build stable structures. This page computes the core trigonometric relationships and explains what each one means.
For a right triangle, the three primary trig functions are defined as ratios of its sides, relative to one of the acute angles. The famous mnemonic SOH-CAH-TOA captures them:
The other three functions are simply the reciprocals of these:
The calculator returns all six for any angle you type in. For example, entering 45° gives sin(45°) = 0.70710678, cos(45°) = 0.70710678, and tan(45°) = 1, along with their reciprocals.
Angles can be measured in two common units, and the calculator supports both. Degrees divide a full circle into 360 equal parts and are the everyday unit most people learn first. Radians measure an angle by the length of the arc it cuts on a unit circle; a full circle is 2π radians, so 180° equals π radians (about 3.14159). Radians are the natural unit in higher mathematics and physics because they make many formulas simpler. To convert, multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. Just pick the unit that matches your problem before you calculate.
A handful of angles come up so often that their trig values are worth knowing by heart:
Notice that tangent is undefined at 90°: its formula divides by the cosine, which is zero there. The calculator reports "undefined" whenever a function would require dividing by zero, such as tan and sec at 90°, or cot and csc at 0°.
A right triangle has one 90° angle. In this calculator, that right angle sits at vertex C, so the side opposite it — side c — is the hypotenuse, the longest side. Side a lies opposite angle α, side b lies opposite angle β, and because the angles of any triangle add to 180°, the two acute angles satisfy α + β = 90°.
To pin down a right triangle completely, you only need two independent pieces of information beyond the right angle itself. The solver accepts any of these combinations:
From your inputs, the calculator finds the missing sides, both acute angles, and the area, using the Pythagorean theorem and the trig ratios above.
The right triangle solver relies on a small set of relationships. With the right angle at C:
By combining these, the tool can start from any valid pair of inputs and rearrange to find everything else.
Suppose a right triangle has legs a = 7 and b = 12. The solver computes:
Every measurement of the triangle falls out of just the two legs you started with.
While SOH-CAH-TOA defines the trig functions for the acute angles of a right triangle, the unit circle extends them to any angle at all — including angles greater than 90° and negative angles. Picture a circle of radius 1 centered at the origin. For any angle θ measured counter-clockwise from the positive x-axis, the point where the angle's ray meets the circle has coordinates (cosθ, sinθ). In other words, cosine is the x-coordinate and sine is the y-coordinate of that point.
This picture explains many properties at a glance. Because the point stays on a circle of radius 1, sine and cosine always fall between −1 and 1. As the angle sweeps around, both functions rise and fall smoothly and repeat every 360° (2π radians) — that is why they are called periodic. The unit circle also shows why sine is positive in the upper half and negative in the lower half, and why cosine is positive on the right and negative on the left. Tangent, being sine over cosine, shoots off to infinity wherever cosine crosses zero (at 90° and 270°). This calculator uses your device's built-in high-precision routines, which are based on exactly these definitions, so it handles any angle you enter, large or small, positive or negative.
The trig functions turn an angle into a ratio; the inverse trig functions do the reverse, turning a ratio back into an angle. They are written arcsin (or sin⁻¹), arccos, and arctan. For example, if you know that the sine of an angle is 0.5, then arcsin(0.5) = 30° tells you the angle. The right triangle solver uses arctangent behind the scenes: when you enter two legs, it computes α = arctan(a ÷ b) to recover the angle from the ratio of the sides.
Inverse functions are essential whenever you know the sides of a triangle and need its angles — a common situation in surveying, navigation, and construction. One subtlety is that inverse functions return a single "principal" value within a limited range (for instance, arcsin returns angles between −90° and 90°), because many different angles can share the same sine or cosine. For the acute angles of a right triangle, though, this is never a problem, since those angles always lie between 0° and 90°.
This calculator focuses on right triangles, but trigonometry also solves triangles with no right angle at all, using two powerful rules. The law of sines states that in any triangle, the ratio of each side to the sine of its opposite angle is the same: a ÷ sin(A) = b ÷ sin(B) = c ÷ sin(C). It is the tool of choice when you know two angles and a side, or two sides and a non-included angle.
The law of cosines generalizes the Pythagorean theorem to any triangle: c² = a² + b² − 2ab·cos(C). When the angle C is 90°, cos(C) is zero and the formula collapses back into the familiar c² = a² + b². The law of cosines is used when you know all three sides, or two sides and the angle between them. Together with the right-triangle relationships on this page, these two laws let you solve virtually any triangle you will meet.
Trigonometric identities are equations that hold true for every angle, and they are the workhorses of algebra and calculus. The most fundamental is the Pythagorean identity: sin²(θ) + cos²(θ) = 1, which follows directly from the unit circle. Others include the co-function identities, such as sin(θ) = cos(90° − θ), which is exactly why the two acute angles of a right triangle have "swapped" sine and cosine values. There are also double-angle formulas like sin(2θ) = 2·sin(θ)·cos(θ), which the solver uses internally when working from an angle and an area. You do not need to memorize these to use the calculator, but knowing they exist helps explain why the numbers relate the way they do.
Trigonometry is one of the most applied areas of mathematics:
In each case, the same handful of ratios — sine, cosine, and tangent — do the heavy lifting.
The tool has two tabs:
Results appear directly below each form, with the key formulas shown so you can follow the working.
They are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). The first three are the primary ratios of a right triangle's sides, and the last three are their reciprocals.
It is a memory aid for the three main trig ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent, all relative to a chosen acute angle in a right triangle.
Multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. For example, 90° = 90 × π/180 = π/2 ≈ 1.5708 radians.
Tangent equals sine divided by cosine, and cos(90°) = 0. Because division by zero is undefined, tan(90°) has no finite value — it grows without bound as the angle approaches 90°.
Two, in addition to the known right angle. That can be two sides, one side and one acute angle, or one side and the area. Two angles alone are not enough because they do not fix the triangle's size.
The hypotenuse is the side opposite the right angle — side c in this calculator. It is always the longest side of a right triangle.
This Trigonometry Calculator is provided for educational and general informational purposes. While it uses the standard trigonometric and right-triangle formulas, you should verify results independently for critical engineering, construction, or academic work.