Tangent Calculator

Tangent is the ratio of the opposite side to the adjacent side — equivalently, sin θ ÷ cos θ. Unlike sine and cosine it isn't bounded between −1 and 1; it runs off to infinity as the angle approaches 90°, because cosine in the denominator heads toward zero. That makes tangent the function that converts a rise-over-run slope directly into an angle, and vice versa.

Worked example — wheelchair ramp gradient

A ramp rises 0.9 m over a 10.8 m run, a gradient of 0.9 ÷ 10.8 = 0.0833. The angle is arctan(0.0833) ≈ 4.76°, just under the commonly cited maximum of about 4.8° (1:12) for accessible ramps. Reversed: a ramp built at exactly 4.76° rises tan(4.76°) × 10.8 ≈ 0.9 m over that same run.

Why it blows up near 90°

At 90°, cos θ is 0, and dividing by zero has no finite answer — tan(90°) is undefined, and values just below it are enormous. This shows up practically as a warning sign: any calculation using tangent near-vertical angles (steep roof pitches, near-vertical camera tilts) is numerically unstable and small input errors produce huge swings in the result.