Trigonometric Function Calculator

Ask a browser for the tangent of 90 degrees and it answers 16331239353195370. Not an error, not infinity, just a sixteen-digit number that looks like a result. It happens because 90° converted to radians is not exactly a quarter turn in binary floating point, so the cosine lands a hair away from zero and the division survives. The Trigonometric Function Calculator decides that case from the angle instead of the float, and says undefined.

Six functions rather than three, since cosecant, secant and cotangent are where the undefined cases actually bite, and any of degrees, radians or gradians going in. Angles that land on the 30/45/60 lattice get their exact surd printed beside the decimal, so 45° reads √2/2 and not only 0.7071067812.

The second mode runs the other way, turning a ratio back into an angle. That direction has a trap of its own, which the page states rather than hides: an inverse function returns one angle out of infinitely many.

Walkthrough: 45 Degrees, Then 90

Load Sample opens on 45 degrees. All six functions are defined: sine and cosine both read 0.7071067812 with √2/2 beside them, tangent and cotangent are exactly 1, and cosecant and secant are 1.414213562. The unit circle draws the angle with its sine and cosine dropped as dashed projections. Press it again for 90 degrees and the table changes character: sine is 1, cosine is 0, and tangent and secant are marked undefined with a note explaining that a calculator working from the floating-point angle prints an enormous number there instead. Keep cycling and you reach pi/3 in radians, 200 gradians (a straight angle, where it is cosecant and cotangent that drop out), and −135°, which normalises into quadrant III.

How It Calculates the Undefined Cases

• The angle is converted to degrees and normalised into 0–360 first. Only then is it checked against the lattice, so a singularity is identified by which angle you asked for, never by how close a float happened to land.
• At 90° and 270° cosine is set to exactly zero, which makes tangent and secant undefined. At 0° and 180° sine is exactly zero, so cosecant and cotangent are the ones that go. Those are the only four singular positions on the circle.
• Radian input accepts multiples of π, and that is not decoration. pi/3 is recognised as 60° and gets exact values; typing 1.0472 does not, because it is near π/3 rather than equal to it. Claiming exactness for an approximation would be the same dishonesty as printing that tangent.
• Gradians are supported because surveying still uses them: a right angle is 100 gradians and a full turn is 400, so 200 gradians is a straight angle.
• Alongside the values you get the normalised angle, the quadrant, and the reference angle, which is the acute angle to the horizontal axis that every value in the table is really a signed copy of.

Accuracy, and What Ten Digits Hides

• Values print to ten significant figures. That is already past where double precision stays trustworthy for very large inputs: normalising a million degrees discards low-order digits before the sine is ever taken, so the last digit or two of such a result is noise.
• Exact surds appear only for genuine lattice angles. Nothing is snapped, rounded into place, or guessed at from a near miss.
• In the inverse direction, arcsine and arccosine reject anything outside −1 to 1 with the reason: sine and cosine never leave that range, so no angle produces the value. Arctangent accepts everything, because tangent does reach every real number.
• The inverse result comes with the whole solution family, not just the principal value. A sine of 0.5 is 30°, and also 150°, and also either of those plus any whole number of turns, which is why a bare 30° is half an answer.
• Negative and over-a-turn angles are fine. −135° is reported as normalising to 225° in quadrant III, and the six values are those of the normalised angle, since the functions repeat every 360°.