Trigonometry Calculator, Explained

After reading this you can evaluate all six trig functions at any angle, read exact values at the special angles, run the inverse functions, and interpret reference angle and quadrant without second-guessing the sign.

What trigonometry measures

Trigonometry connects an angle to a ratio of lengths. Draw a circle of radius 1 centered at the origin. Pick an angle \theta measured counterclockwise from the positive x-axis. The point where that angle meets the circle has coordinates (\cos\theta, \sin\theta). So cosine is the x-coordinate, sine is the y-coordinate, and both stay between -1 and 1.

Here is the hook. At \theta = 30^\circ the point is (\tfrac{\sqrt{3}}{2}, \tfrac{1}{2}). That means \sin 30^\circ = 0.5 exactly, and \cos 30^\circ = 0.8660. No measurement needed. The unit circle hands you the value.

The remaining four functions are ratios built from those two:

\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \csc\theta = \frac{1}{\sin\theta}

Tangent is the slope of the radius line. Cotangent, secant and cosecant are the reciprocals of tangent, cosine and sine. At 30^\circ, \tan 30^\circ = 0.5 / 0.8660 = 0.5774.

When to use this calculator

Use it whenever you need a single value: a function at a fixed angle, or an angle recovered from a known ratio. It gives exact forms like \tfrac{\sqrt{3}}{2} at the special angles and full decimal precision everywhere else. It also reports the reference angle and quadrant, which are the two facts that determine sign.

It is not a triangle solver. If you have side lengths and want the missing sides and angles, use the Triangle Solver or the Right Triangle Calculator. For arc length or sector area on a circle, use the Circle Calculator. This page evaluates functions, it does not fit them to a figure.

Degrees, radians, and why the unit matters

An angle can be measured in degrees or radians. A full turn is 360 degrees or 2\pi radians, so the conversion is:

\theta_{\text{rad}} = \theta_{\text{deg}} \cdot \frac{\pi}{180}

Each degree equals \pi/180 \approx 0.01745 radians. So 30^\circ = 0.5236 radians. This matters because the calculator trusts the unit you choose. Feed it 30 in radians and it treats that as almost five full turns, giving \sin(30\ \text{rad}) = -0.9880 rather than 0.5.

The single most common trig error is a unit mismatch. If your answer for \sin 30 comes out near -0.988 instead of 0.5, you asked for radians when you meant degrees. Check the unit before you trust the number.

Reference angle and quadrant

Every angle reduces to a first-quadrant twin called the reference angle, the acute angle between the terminal side and the x-axis. The function values match that twin in size; only the sign changes with the quadrant.

Signs of sine, cosine and tangent by quadrant
Quadrant	Angle range	sin	cos	tan
I	0 to 90	+	+	+
II	90 to 180	+	-	-
III	180 to 270	-	-	+
IV	270 to 360	-	+	-

Take \theta = 210^\circ. It sits in quadrant III, and its reference angle is 210 - 180 = 30^\circ. So \sin 210^\circ has the same magnitude as \sin 30^\circ = 0.5 but a negative sign: \sin 210^\circ = -0.5. Cosine is also negative there, -0.8660, while tangent is positive, 0.5774.

Worked example: sine of 30 degrees

Reproducing the demo

Set Mode to Evaluate at an angle, Angle to 30, and Unit to Degrees, then press Calculate.

Locate the angle. 30^\circ is in quadrant I, so all six functions are positive. The reference angle equals the angle itself, 30^\circ.
Read the unit-circle point: (\cos 30^\circ, \sin 30^\circ) = (\tfrac{\sqrt{3}}{2}, \tfrac{1}{2}).
State the exact and decimal values. \sin 30^\circ = \tfrac{1}{2} = 0.5 and \cos 30^\circ = \tfrac{\sqrt{3}}{2} = 0.8660.
Divide for tangent: \tan 30^\circ = 0.5 / 0.8660 = \tfrac{1}{\sqrt{3}} = 0.5774.
Invert for the reciprocals: \csc 30^\circ = 2, \sec 30^\circ = 1.1547, \cot 30^\circ = \sqrt{3} = 1.7321.

Every value here is exact. That is the payoff of the special angles: no rounding creeps in until you round on purpose.

Sine climbs to 1 at 90 degrees, returns to 0 at 180, dips to -1 at 270. The marker at 30 degrees sits at 0.5.

The inverse functions

The inverse mode runs the process backward: you give a ratio, it returns an angle. The three inverses answer these questions.

arcsin: The angle whose sine is the input. Accepts -1 to 1. Returns an angle in [-90^\circ, 90^\circ].
arccos: The angle whose cosine is the input. Accepts -1 to 1. Returns an angle in [0^\circ, 180^\circ].
arctan: The angle whose tangent is the input. Accepts any real number. Returns an angle in (-90^\circ, 90^\circ).

Set Function to arcsin and Value to 0.5. The result is \arcsin 0.5 = 30^\circ, the exact inverse of the worked example above. Try arccos with 0.5 and you get 60^\circ, since \cos 60^\circ = 0.5.

Inverses return one angle, but many angles share a value. Because \sin 30^\circ = \sin 150^\circ = 0.5, arcsin picks only the one in its range, 30^\circ. If your problem lives in quadrant II, add the second solution yourself: 180^\circ - 30^\circ = 150^\circ.

Explore how the angle moves the point

Common mistakes

Beyond the unit mismatch already flagged, three errors recur.

Feeding the inverse an out-of-range value. Since sine and cosine never exceed 1, \arcsin 1.5 has no real answer. The calculator will reject it. Arctangent has no such limit; it accepts any number and returns an angle approaching 90^\circ as the input grows.

Forgetting that tangent, secant, cotangent and cosecant blow up. Tangent is undefined at 90^\circ and 270^\circ because cosine is 0 there. Near 89^\circ tangent is already 57.29, and it races to infinity. A huge tangent is not a bug; you are near a vertical asymptote.

Dropping the sign from the reference angle. The reference angle is always positive and acute. The actual value carries the quadrant sign. Compute the magnitude from the reference angle, then attach the sign from the quadrant table.

Tangent stays modest through 60 degrees, then climbs steeply. By 89 degrees it exceeds 57.

Related tools

Once you have the ratios, other pages turn them into shapes and measurements. Solve full triangles with the Triangle Solver. Work coordinates, slopes and intersections with the Coordinate Geometry Toolkit. Find areas from vertices with the Polygon Area Calculator, or the angles and radii of a regular shape with the Regular Polygon Calculator. For volumes and surface areas, use the 3D Solid Geometry Calculator.

Frequently asked questions

Why does sin 30 give -0.988 on my other calculator?

That calculator is in radian mode. It reads 30 as 30 radians, which is 30 - 4(2\pi) \approx 4.867 radians past the start. Switch to degrees to get the expected 0.5.

What are the special angles?

They are the angles with clean exact values: 0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ and their reflections around the circle. For example \cos 45^\circ = \tfrac{\sqrt{2}}{2} = 0.7071 and \sin 60^\circ = \tfrac{\sqrt{3}}{2} = 0.8660.

Why does arcsin only return angles from -90 to 90 degrees?

Sine repeats, so a single value maps to infinitely many angles. To make arcsin a genuine function it is restricted to one interval where sine is one-to-one. That interval is [-90^\circ, 90^\circ]. Add or reflect to recover other solutions.

How do I get cosine or tangent of an inverse?

Run the inverse to get the angle, then evaluate the second function at that angle. For instance \cos(\arcsin 0.5) = \cos 30^\circ = 0.8660. You can also use the identity \cos(\arcsin x) = \sqrt{1 - x^2}, which gives \sqrt{1 - 0.25} = 0.8660.

What is the reference angle of 210 degrees?

It is 30^\circ. Angle 210^\circ sits in quadrant III, so subtract 180^\circ. The function magnitudes match 30^\circ, with signs taken from quadrant III.

Try it yourself in the tool