The word “tangent” comes from “tangens”, meaning touching or extending (the line that touches the circle at one point).
The term “tangent” referring to an angle was first used by the Danish mathematician Thomas Fincke in 1583. It derives from the fact that the tangent of an angle α (oriented as in the figure, so that the first side coincides with the semi-axis of the positive x) can be defined (if it exists) as the ordinate of the point Q of intersection between its second side and the tangent line to the trigonometric circle (circle with center O and unit radius) at the point of coordinates (1,0) (see the figure).
Previously, instead of the word tangent, the term “umbra recta” (straight shadow) was used in reference to the use of the tangent function in problems related to calculating the length of a shadow cast by an object.
TOA (Tangent is the opposite over the adjacent)
We can state a more general definition of tangent using a right triangle.
Note that this definition of tangent is equivalent to the definition above since.
The Graph of the Tangent Function
We can sketch a graph of the tangent function by creating a table of input and output values and plotting these points on a plane. Below is a table of values for f(θ) = tan(θ) and its corresponding graph.
Notice that the output values of the tangent repeat on a regular interval, so f(θ) = tan(θ) is a periodic function. For any angle, there is a second angle halfway around the unit circle with the same tangent value. Therefore, the period of tangent is π.π. We can see one continuous cycle from −π/2 to π/2, before the graph jumps and repeats itself.
References
Brummer, Josh and Taran Funk, et al. 2025. “Mathematics: PreCalculus Mathematics at Nebraska.” MFG The Tangent Function. Accessed January 10. https://mathbooks.unl.edu/PreCalculus/tangent-and-cofunctions.html.
“The term “tangent” referring to an angle was first used by the Danish mathematician Thomas Fincke in 1583.” 2025. Quora. Accessed January 10. https://qr.ae/pYYmkh.
BLUE, THE TRIGONOGRAPHER. “(ALMOST) EVERYTHING YOU NEED TO REMEMBER ABOUT TRIGONOMETRY, IN ONE SIMPLE DIAGRAM.” 2025. Accessed January 10. https://tricochet.com/math/pdfs/AlmostEverythingAboutTrig.pdf.
The featured image on this page is from the Mathematics: PreCalculus Mathematics at Nebraska website.
Mathematical Mysteries 