Right-Triangle Tangent Ratio (TOA)

TOA — Tangent is Opposite over Adjacent. It is the ratio to reach for when the hypotenuse is unknown, and in the field the hypotenuse usually is unknown, because it runs along your line of sight to something you cannot put a tape on. Tangent is not independent of the other two: dividing $\sin\theta = o/h$ by $\cos\theta = a/h$ cancels the hypotenuse and leaves $o/a$, which is the formula on this page and the reason the hypotenuse drops out of the problem.

Here is the fact that makes this the most quietly useful ratio of the three: tangent is slope. Rise over run is opposite over adjacent, so $m = \tan\theta$, and every grade, pitch and gradient you have ever seen is a tangent in disguise. A 6% road grade is $\arctan(0.06) = 3.43°$. A 6:12 roof pitch is $\arctan(0.5) = 26.6°$. A 1:12 ramp is 4.76°. If you have ever wondered why a highway sign warning of an 8% grade describes something that looks nowhere near eight degrees, this is why — and it is worth noticing that a 100% grade is 45°, not vertical.

The classic use is an angle of elevation. Stand 50 m out from a tower on level ground and sight the top at 31°: the rise above your eye is $50\tan 31° = 30.0$ m. Backwards, $\theta = \arctan(o/a)$, and the principal branch is exactly right here — two positive legs always land in 0°–90°, so there is no ambiguity to resolve.

Four cautions, in the order they cost people money. First, that tower is not 30 m tall. The formula returns the rise above the instrument, so you must add your eye height, or the tripod height, to get the real total — a systematic error of a metre and a half that stays invisible because the answer looks reasonable. Second, $a$ is the horizontal distance. On sloping ground a taped distance runs along the slope and is longer than the run, which inflates the result; that is why survey instruments reduce slope distance to horizontal before anything else happens. Third, tangent has no ceiling — it runs to infinity at 90° — and grows viciously sensitive as it approaches. At 30° a half-degree error in the sighting shifts the height by about 2%; at 80° the same half degree shifts it by 5%, and at 85° by 10%. Standing farther back and sighting at a shallower angle is nearly always the more accurate measurement. Fourth, as ever, check whether your calculator is in degrees.