Tangent Calculator

What is Tangent?

The tangent of an angle θ is the ratio of the opposite side to the adjacent one: tan(θ) = sin(θ) / cos(θ). Unlike sine and cosine, it has no bound and can take any real value.

Among the notable values are tan(0°) = 0, tan(30°) ≈ 0.5774, tan(45°) = 1 and tan(60°) ≈ 1.7321. At 90°, 270° and the other odd multiples of 90°, tangent becomes undefined.

Unit conversions: 1° equals π/180 rad ≈ 0.01745 rad, while 1 rad equals 180°/π ≈ 57.296°. And 400 gradians correspond to 360°.

The tangent function

The tangent is defined as tan(θ) = sin(θ)/cos(θ), which in a right triangle equals the ratio opposite side / adjacent side. It has period π (half that of sine and cosine) and vertical asymptotes wherever the cosine vanishes, i.e. at θ = π/2 + kπ. It is an odd function: tan(−θ) = −tan(θ). Reference values: tan(0) = 0, tan(30°) = √3/3, tan(45°) = 1, tan(60°) = √3, and tan(90°) undefined. Its derivative is (tan x)' = sec²x.

Applications: slopes, roofs and accessibility

The tangent is the natural way to express slope: a ramp with angle α has tan(α) = rise / run. The Brazilian accessibility standard NBR 9050 caps ramps at 8.33% (about 4.76°). Roofers use it to convert pitch to angle, surveyors measure building heights by reading the angle of elevation to the top and applying height = distance · tan(α), optics uses it in Snell's law and Brewster's angle, and GPS triangulation relies on tangent ratios between known points.