By Lee Johnson | Updated Aug 30, 2022
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Trigonometry is more than a set of obscure symbols—it's a powerful tool that underpins many scientific and engineering disciplines. Understanding how to translate a tangent value into a familiar degree measure unlocks practical applications, from navigation to structural analysis.
For a right‑angled triangle, tan θ = opposite / adjacent. To convert a tangent value back to a degree measure, use the inverse function: θ = arctan(tan θ), which on most calculators appears as tan⁻¹.
In a right‑angled triangle, the tangent of an angle θ is the ratio of the side opposite that angle to the side adjacent to it:
\(\tan(\theta) = \dfrac{\text{opposite}}{\text{adjacent}}\)
Because the tangent relies only on the two legs of the triangle, the hypotenuse plays no role in its calculation. Alternatively, tan θ can be expressed as the ratio of sine to cosine:
\(\tan(\theta) = \dfrac{\sin(\theta)}{\cos(\theta)}\)
The inverse tangent, or arctan (often written as tan⁻¹), undoes the tan operation. If you know tan θ, applying arctan returns the original angle θ, expressed in radians or degrees depending on your calculator’s settings. Arcsin and arccos perform the same reverse operations for sine and cosine, respectively.
To find an angle in degrees from a given tangent value, simply apply the arctan function:
\(\text{Angle in degrees} = \arctan(\tan(\theta))\)
For example, if tan θ = √3, then:
\(\begin{aligned} \text{Angle in degrees} &= \arctan(\sqrt{3})\\ &= 60^\circ \end{aligned}\)
On most calculators, press the tan⁻¹ button before entering the value, or after, depending on the model.
Consider a boat traveling eastward at 5 m/s while a northward current pushes it at 2 m/s. What is the resultant direction relative to due east?
Model the situation as a right‑angled triangle: the eastward speed is the adjacent side, the northward current is the opposite side, and the combined velocity is the hypotenuse. Thus:
\(\tan(\theta) = \dfrac{2\,\text{m/s}}{5\,\text{m/s}} = 0.4\)
Converting to degrees:
\(\begin{aligned} \text{Angle in degrees} &= \arctan(0.4)\\ &\approx 21.8^\circ \end{aligned}\)
The boat’s trajectory deviates 21.8° north of east, illustrating how tangent values translate directly into navigational bearings.
