Consider a right triangle with a hypotenuse that measures 1 unit, or a trigonometric circle in which r = 1.
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In this right triangle, we have the relations: \(\sin \left( \theta \right) = y\) and \(\cos \left( \theta \right) = x\).
Therefore, \(\tan \left( \theta \right) = \dfrac{\sin \left( \theta \right)}{\cos \left( \theta \right)} = \dfrac {y}{x}\)
Notation
The notation used to indicate the tangent of a real number x is “tan(x)” which is read as “the tangent of x.”
Educational Note
It should be noted that the argument of the tangent is a number (a measurement) and not a geometric figure (an angle). It’s a linguistic shortcut to use the expression “tangent of an angle” to express the “tangent of the measure of an angle.”