Trigonometric Functions Calculator — all six functions, exact and decimal

This trigonometric functions calculator evaluates all six circular functions for a single angle at once: sine, cosine, tangent, and their three reciprocals cosecant, secant and cotangent. It accepts degrees or radians, reports the angle in both units so you never have to convert by hand, names the quadrant the angle lands in, and prints the exact surd value — such as sin 60° = √3⁄2 ≈ 0.8660 — whenever the angle is one of the standard ones a textbook expects you to know by heart.

Arb Digital builds and maintains a large free calculator library, and the trigonometry pages were written after watching how often a phone calculator hands back a confusing answer: a tiny number like 6.1×10^-17 where the true value is zero, or a screen full of digits where the answer should be a clean fraction. This tool is deliberately honest about both problems. Where a function is genuinely undefined, it says undefined and tells you why, instead of printing an astronomically large number and letting you assume it is real.

What This Trigonometric Functions Calculator Does

Give it one angle and it returns seven pieces of information. The headline result is sin θ. The supporting grid gives cos θ, tan θ, csc θ, sec θ and cot θ, then the same angle expressed in the unit you did not choose, plus the quadrant. The quadrant matters more than most people expect, because it is the single fact that determines the sign of every one of the six results. Once you know an angle sits in the third quadrant, you already know sine is negative, cosine is negative and tangent is positive, before any arithmetic happens.

The exact-value line is the part that separates this from a phone calculator. At every multiple of 15° the sine and cosine have closed forms built from square roots, and the calculator prints them. sin 45° is √2⁄2, not merely 0.7071. cos 150° is −√3⁄2. tan 15° is 2 − √3. Those forms are what an exam answer sheet wants, and they are also what lets you carry a value through an algebraic simplification without accumulating rounding error.

How to Use It

1. Type the angle. Any real number works, including negatives such as −120 and values beyond a full turn such as 780.
2. Pick the unit. Degrees is selected by default because most people arrive here from a geometry problem. Switch to radians if your angle came from calculus, physics, or a programming language, where radians are the standard.
3. Set the decimal places. Four is a sensible default; raise it if you are checking a value against a table, lower it for a tidy answer.
4. Press Calculate. All six functions update together, along with the quadrant and the converted angle.
5. Read the tip line. When the angle is a standard one, the exact surd values for sine, cosine and tangent appear there.

The Formula: How Each Function Is Calculated

Every one of the six functions is defined from a single point on the unit circle — the circle of radius 1 centred on the origin. Rotate anticlockwise from the positive x‑axis by the angle θ, and the point where you land has coordinates (cos θ, sin θ). That is the whole definition. Cosine is the horizontal coordinate, sine is the vertical coordinate, and everything else is built from those two:

tan θ = sin θ ÷ cos θ, csc θ = 1 ÷ sin θ, sec θ = 1 ÷ cos θ, and cot θ = cos θ ÷ sin θ. The reciprocal pairings are the ones students most often mix up: cosecant is the reciprocal of sine, and secant is the reciprocal of cosine, even though the leading "co−" makes the opposite pairing look natural. Our unit circle calculator shows the same relationships laid out geometrically, and the Pythagorean theorem calculator covers the identity sin²θ + cos²θ = 1 that follows directly from the circle having radius 1. The NIST Digital Library of Mathematical Functions states these definitions and their periodicity formally.

Where the Functions Are Undefined — and Why Calculators Lie About It

Four of the six functions have angles where they simply do not exist. Tangent and secant are undefined wherever cos θ = 0, which is 90°, 270°, and every 180° step from there. Cosecant and cotangent are undefined wherever sin θ = 0, which is 0°, 180°, 360° and so on. These are not awkward edge cases; they are vertical asymptotes, points where the function grows without bound in one direction and plunges without bound from the other side.

Here is the practical problem. Because computers store π as a finite decimal, cos(90°) inside a floating-point library does not come out as exactly zero — it comes out as roughly 6.1×10^-17. Divide 1 by that and you get about 1.6×10¹⁶, a colossal but finite number. Many calculators print it. A student then writes down "tan 90° ≈ 1.6×10¹⁶", which is wrong: tan 90° does not have a value at all. This calculator checks whether the cosine or sine is within a tiny tolerance of zero and, if it is, prints undefined together with the reason. That single behaviour prevents a whole family of downstream errors, particularly in physics problems where an undefined tangent usually signals that the geometry itself has broken down — a ray parallel to a surface, or a slope that is genuinely vertical.

Why the Same Angle Gives Two Different Answers

The most common wrong answer in trigonometry is not a mis-typed formula. It is a calculator left in the wrong angle mode. Ask for sin(30) with the mode set to radians and you get about −0.988; ask with degrees and you get exactly 0.5. Both are correct answers to different questions, and nothing on the screen warns you which question was asked. That is why the unit selector here sits directly under the angle field rather than buried in a settings menu, and why the results panel always echoes the angle back in both units. If the degrees figure and the radians figure do not look like the angle you meant, you have caught the error before it propagates.

Radians are not an arbitrary alternative to degrees. A radian is the angle that subtends an arc equal in length to the radius, which makes the arc-length formula simply s = rθ with no conversion constant, and makes the derivative of sin x equal cos x exactly. Degrees carry a factor of π⁄180 through every one of those results. If you only need to move a number between the two systems rather than evaluate a function, our angle converter handles degrees, radians and gradians directly.

Reading Signs From the Quadrant Instead of Memorising Them

Signs follow from the coordinates, so there is nothing to memorise if you can picture the circle. In the first quadrant both coordinates are positive, so all six functions are positive. In the second quadrant the x‑coordinate turns negative while y stays positive: sine and its reciprocal cosecant stay positive, everything else flips negative. In the third quadrant both coordinates are negative, so sine and cosine are both negative but tangent — being their ratio — is positive again, along with cotangent. In the fourth quadrant x is positive and y is negative, leaving cosine and secant positive.

The traditional mnemonic compresses this into four letters, one per quadrant: All, Sine, Tangent, Cosine, naming which functions are positive in each quadrant starting from the first and going anticlockwise. It works, but it is worth knowing that it is a summary of the coordinate picture rather than a rule in its own right. The reference angle calculator handles the other half of this job: it strips an angle down to its acute equivalent so you can look up the magnitude, then you apply the quadrant sign yourself.

Exact Values, Surds, and Why 0.8660 Is Not the Answer

At the standard angles the trigonometric functions take values expressible with square roots, and those forms are exact where a decimal is only an approximation. The core set is small enough to memorise: sin 30° = 1⁄2, sin 45° = √2⁄2, sin 60° = √3⁄2, with cosine reading the same list backwards. The half-step angles come from the sum formulas — sin 15° = (√6 − √2)⁄4 is what you get by expanding sin(45° − 30°).

Keeping the surd form matters whenever the value is not the final answer. Square √3⁄2 and you get exactly 3⁄4; square 0.8660 and you get 0.74995, which then contaminates every later step. In a long derivation those small deviations accumulate until a result that should be a clean integer arrives as 5.9998. The rule that professional practice follows is simple: carry exact values as far as you can, and convert to decimals only at the last step, at the precision the application actually needs. Wolfram MathWorld's trigonometry entry catalogues the closed forms in more depth than any single page can summarise.

Periodicity: Why 780° and 60° Give Identical Results

Sine and cosine repeat every 360° (2π radians), because 360° brings you back to the same point on the circle. So sin 780° = sin 60°, since 780 − 720 = 60. Tangent and cotangent are stranger: they repeat every 180°, not 360°, because the ratio of the two coordinates is unchanged when both flip sign. tan 200° equals tan 20° exactly.

That shorter period is the reason inverse tangent behaves so differently from inverse sine, and it is worth understanding before you use any of the inverse tools. If you need the angle back from a ratio rather than the ratio from an angle, use the arcsin calculator, arccos calculator or arctan calculator — each returns one principal value, and each has a different range because of exactly this periodicity. The coterminal angle calculator is the direct tool for finding all the angles that share a terminal side.

Common Mistakes to Avoid

• Leaving the calculator in the wrong angle mode — sin 30 in radians is about −0.988, nowhere near the 0.5 you expected. Always check the unit before trusting a result.
• Treating a huge number as tan 90° — tangent has no value there. A result like 1.6×10¹⁶ is a floating-point artefact, not mathematics.
• Swapping cosecant and secant — csc is 1÷sin and sec is 1÷cos, which is the opposite of what the prefixes suggest.
• Rounding to four decimals mid-derivation — keep exact surds until the final step or the error compounds.
• Assuming tangent repeats every 360° — it repeats every 180°, which changes how many solutions an equation has.

Related Free Tools From Arb Digital

Pair this page with the unit circle calculator for coordinates and quadrant signs, the double angle calculator when your argument is 2θ, and the law of sines calculator or law of cosines calculator when you are solving a whole triangle rather than evaluating one angle. For general arithmetic there is the scientific calculator, and the full free online tools hub lists everything else.

Frequently Asked Questions

This tool is provided for education and reference. Verify any result that will be used in engineering, surveying or safety-critical work against an independent method.