Tangent

Definition

Definition

Let $x \in \mathbb R$, we define the tangent function as

$$\tan(x) = \frac{\sin(x)}{\cos(x)}.$$

Results

Proposition : Derivative

Let $x \in \mathbb R$, we have

$$\tan'(x) = 1 - \tan^2(x) = \frac{1}{\cos^2(x)}.$$

Proposition : Addition formula

Let $a,b \in \mathbb R$ such that $a \neq \frac{\pi}{2}[\pi]$ and $b \neq \frac{\pi}{2}[\pi]$, we have

$$\tan(a+b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}.$$

Proposition : Outstanding values

Some outstanding values to keep in mind

$$\tan(0) = 0, \quad \tan \left (\frac{\pi}{6} \right ) = \frac{\sqrt{3}}{3}, \quad \tan \left (\frac{\pi}{4} \right ) = 1, \quad \tan \left (\frac{\pi}{3} \right ) = \sqrt 3.$$