How this instrument works
Tangent starts as a ratio: tan(θ) = sin(θ) ⁄ cos(θ), the vertical rise divided by the horizontal run of the same point that defines sine and cosine on a unit circle. In a right triangle it is the opposite side over the adjacent side, the ratio surveyors use to turn a measured angle into a height or a grade. The name is not a coincidence: draw the actual line tangent to the unit circle at the point (1, 0), extend the angle's ray until it crosses that tangent line, and the length of the crossing segment is tan(θ) itself — geometry gave the function its name centuries before anyone wrote sin(θ) ⁄ cos(θ) as a formula.
Because cosine sits in the denominator, tangent inherits none of sine's or cosine's neat boundaries. As θ climbs toward 90°, cosine shrinks toward zero while sine stays near 1, so the ratio grows without limit — tan(89.9°) already exceeds 500, and at exactly 90° the function is undefined, a vertical asymptote rather than a missing rounding digit. The same blow-up recurs every 180°, at 90°, 270°, 450°, and onward, anywhere cosine crosses zero.
That 180° spacing is tangent's other structural difference from sine and cosine: while they need a full 360° turn to repeat, tangent repeats after only half a turn, because flipping both sine and cosine's signs at once — exactly what adding 180° does — leaves their ratio unchanged. Tangent is also an odd function, tan(−θ) = −tan(θ), and it carries a second life as a slope: the tangent of the angle a line makes with the horizontal is precisely that line's rise over run, which is why a 45° ramp climbs at a one-to-one grade.
- Enter your angle into the Angle, θ field; the unit selector opens on degrees, so a plain 45 means 45°.
- Switch to radians or turns if that's how your angle was measured — the sheet converts before evaluating anything.
- Read tan(θ) for the result, and watch for very large numbers as θ nears 90°, 270°, or any multiple of 90° plus 180°.
- If the field reports an error instead of a number, θ has landed exactly on 90° (or 270°, 450°…), where cosine is zero and tangent is undefined.
- Step θ through 0°, 45°, and close to 90° to watch tan(θ) move from 0 to 1 and then run away toward infinity.
Worked example — the 45° reference angle
Set the Angle, θ field to 45 with the unit selector on degrees. Internally the sheet converts that to 0.7853981633974483 radians — exactly π ⁄ 4 — before evaluating, and tan(θ) returns 0.9999999999999999, a floating-point hair below the true value of 1 that the display rounds cleanly to 1.0. At 45° the point on the unit circle sits exactly on the line y = x, so its rise and its run are identical and their ratio is exactly 1.
The result checks out two ways. Since sin(45°) and cos(45°) both equal √2⁄2 ≈ 0.7071067811865476, dividing one by the other cancels the shared factor entirely and leaves 1. And because tan(θ) equals rise over run, a ramp built at exactly 45° climbs one metre for every metre it advances — the steepest angle where climbing and advancing cost exactly the same.
Questions
What does tan(θ) actually measure?
It's sine divided by cosine — the ratio of a unit-circle point's y-coordinate to its x-coordinate, or equivalently the opposite side over the adjacent side in a right triangle. Unlike sine or cosine, which are themselves single coordinates, tangent is built from two other numbers, which is exactly why it inherits none of their bounded, well-behaved range.
Why is tan(θ) undefined at 90°?
Because tan(θ) = sin(θ) ⁄ cos(θ), and cos(90°) = 0 — dividing by zero has no defined result, so the function has a vertical asymptote there instead of a value. Just before 90° the ratio grows past any bound you name, and just after it flips to a large negative number, which is what a genuine asymptote looks like rather than a rounding artifact.
Why does tangent repeat every 180° instead of 360° like sine and cosine?
Adding 180° to an angle flips the sign of both sine and cosine — sin(θ+180°) = −sin(θ) and cos(θ+180°) = −cos(θ) — but their ratio cancels the two minus signs, leaving tan(θ+180°) = tan(θ) unchanged. Sine and cosine individually need a full 360° turn to return to their starting value, but tangent, built from their ratio, gets back to the same number after only half that turn.
How does tan(θ) relate to the slope of a line?
They're the same number. If a straight line makes angle θ with the horizontal, its slope — rise divided by run — equals tan(θ) exactly, since rise and run are just the vertical and horizontal legs of the right triangle the line forms with the x-axis. A 45° line has slope 1; a line steeper than 45° has a slope, and a tangent, greater than 1.
What is the origin of the name 'tangent'?
It comes from an actual tangent line to a circle. Draw a unit circle and a vertical line touching it at (1, 0); extend the ray for angle θ from the center until it crosses that tangent line, and the length of the crossing segment is tan(θ). Trigonometers named the ratio after this literal geometric line centuries before sin(θ) ⁄ cos(θ) became the standard definition.
What's the most common mistake when computing tan(θ) by hand?
Feeding a calculator degrees when it's set to radian mode, or the reverse. tan(45) in degree mode is 1; the same keystrokes in radian mode compute the tangent of 45 radians, a value near 1.6197, since 45 radians is more than seven full turns around the circle. This sheet asks which unit you mean and converts before evaluating anything.