How To Convert Tangents To Degrees

The mere mention of the word trigonometry might send a shiver down your spine, evoking memories of high school math classes and arcane terms like sin, cos and tan that never quite seemed to make sense. But the truth is that trigonometry has a huge range of applications, particularly if you're involved in science or math as part of your continuing education. If you're unsure what a tangent really means or how you extract useful information from it, learning to convert tangents to degrees introduces the most important concepts.

In trigonometry, the tangent of an angle can be found using the lengths of the sides of a right-angled triangle containing the angle. The adjacent side sits horizontally next to the angle you're interested in, and the opposite side stands vertically, opposite the angle you're interested in. The remaining side, the hypotenuse, has a part to play in the definitions of cos and sin but not of tan.

With this generic triangle in mind, the tangent of the angle (θ) can be found using:

\(\tan (θ) = \frac{\text{opposite}}{\text{adjacent}}\)

Here, opposite and adjacent describe the lengths of the sides given those names. Thinking about the hypotenuse as a slope, the tan of the angle of the slope tells you the rise of the slope (i.e., the vertical change) divided by the run of the slope (the horizontal change).

The tan of an angle can also be defined as:

\(\tan (θ) = \frac{\sin(θ)}{\cos(θ)}\)

The tangent of an angle technically tells you what the tan function returns when you apply it to the specific angle you have in mind. The function called "arctan" or tan⁻¹ reverses the tan function, and returns the original angle when you apply it to the tan of the angle. Arcsin and arccos do the same thing with the sin and cos functions, respectively.

Converting tangents to degrees requires you to apply the arctan function to the tan of the angle you're interested in. The following expression shows how to convert tangents to degrees:

\(\text{Angle in degrees} = \arctan (\tan (θ))\)

Simply put, the arctan function reverses the effect of the tan function. So if you know that tan (θ) = √3, then:

\(\begin{aligned}
\text{Angle in degrees} &= \arctan (\sqrt{3}) \
&= 60°
\end{aligned}\)

On your calculator, press the "tan⁻¹" button to apply the arctan function. You either do this before entering the value you want to take the arctan of or after, depending on your specific model of calculator.

The following problem illustrates the usefulness of the tan function. Imagine somebody traveling at 5 meters per second in the east direction (from the west) on a boat, but traveling in a current pushing the boat towards the north at 2 meters per second. What angle does the resulting direction of travel make with due east?

Break the problem down into two parts. First, the travel towards the east can be considered to form the adjacent side of a triangle (with a length of 5 meters per second), and the current moving to the north can be considered to be the opposite side of this triangle (with a length of 2 meters per second). This makes sense because the final direction of travel (which would be the hypotenuse on the hypothetical triangle) results from the combination of the effect of the motion towards the east and the current pushing to the north. Physics problems often involve creating triangles like this, so simple trigonometry relationships can be used to find the solution.

Since:

\(\tan (θ) = \frac{\text{opposite}}{\text{adjacent}}\)

This means that the tan of the angle of the final direction of travel is:

\(\begin{aligned}
\tan (θ) &= \frac{2 \text{ m/s}}{5\text{ m/s}} \
&= 0.4
\end{aligned}\)

Convert this to degrees using the same approach as in the previous section:

\(\begin{aligned}
\text{Angle in degrees} &= \arctan (\tan (θ)) \
&= \arctan (0.4) \
&= 21.8°
\end{aligned}\)

So the boat ends up traveling in a direction 21.8° out from the horizontal. In other words, it still moves largely towards the east, but it also travels slightly north because of the current.