How this instrument works
The tangent of an angle, tan(x), is defined as sine divided by cosine: tan(x) = sin(x) ⁄ cos(x). In a right triangle, this is exactly the ratio of the side opposite a given angle to the side adjacent to it — the TOA step in the SOH-CAH-TOA mnemonic, and geometrically, it's the slope of the line from the origin to the corresponding point on the unit circle.
Unlike sine and cosine, which stay confined between −1 and 1 no matter the angle, tangent is unbounded — it grows without limit as the angle approaches 90°, where cosine (its denominator) shrinks toward zero. This is exactly why tangent, unlike its two relatives, has vertical asymptotes rather than a smooth, bounded wave shape.
Tangent shows up constantly outside pure trigonometry: a road or roof's PITCH (rise over run) is literally a tangent value, and a line's SLOPE in coordinate geometry is the tangent of the angle that line makes with the horizontal axis — the same ratio, computed the same way, wearing different practical names.
- Enter an angle into the Angle field.
- Read tan(x): the sheet applies sin(x) ⁄ cos(x) directly.
- Try an angle close to 90° to see the result grow rapidly toward that vertical asymptote.
Worked example — tan(45°)
tan(45°) = 1 exactly — at 45°, a right triangle's opposite and adjacent sides are equal by definition (an isosceles right triangle), making their ratio exactly 1, a clean value worth recognizing immediately.
tan(0°) = 0 exactly, since there's no opposite side at all at this trivial reference angle. tan(60°) = √3 ≈ 1.732 instead — one of the handful of clean values tied to the 30-60-90 triangle's fixed side ratios.
Questions
What is the tangent of an angle?
tan(x) = sin(x) ⁄ cos(x) — in a right triangle, this equals the side opposite the angle divided by the side adjacent to it, the TOA step in SOH-CAH-TOA.
Why is tangent unbounded, unlike sine or cosine?
Because tangent is a ratio with cosine in the denominator, and cosine shrinks toward zero as the angle approaches 90° — dividing by a number approaching zero sends the ratio toward infinity, unlike sine and cosine themselves, which both stay confined between −1 and 1.
What is tangent at exactly 90°?
Undefined — cosine equals exactly zero at 90°, and dividing by zero has no defined result. Tangent has a vertical asymptote at this angle, growing without bound as the angle approaches it from either side.
How is tangent related to slope?
Directly — a line's slope in coordinate geometry equals the tangent of the angle that line makes with the horizontal axis. This is the same reason a road or roof's pitch, quoted as a rise-over-run ratio, is itself a tangent value.
Is tangent the same as inverse tangent?
No, they're opposite operations — tangent takes an angle and returns a ratio, while inverse tangent (atan) takes a ratio and returns the angle that produced it. This page computes the forward direction; the companion Inverse Tangent page handles the reverse.