Real Tangent Function

The Real Tangent Function

Definition: Real Tangent Function

The real tangent function is defined as the ratio of the Real Sine Function to the Real Cosine Function.

$$\tan (x)\stackrel{\text{def}}{=}\frac{\sin (x)}{\cos (x)}$$

NOTE

The domain of the Real Tangent Function is $\{x\in \mathbb{R}\mid x\ne \frac{\pi }{2}+k\pi ,k\in \mathbb{Z}\}$ because $\cos (\frac{\pi }{2}+k\pi )=0$ for all $k\in \mathbb{Z}$.

NOTATION

$\tan (x)\mathrm{tg}(x)$

Properties

Theorem: Image of the Real Tangent Function

The image of the Real Tangent Function is $(-\infty ;+\infty )$.

PROOF

TODO

Theorem: Periodicity of the Real Tangent Function

The Real Tangent Function has a period of $\pi$. More generally,

$$\tan (x+k\pi )=\tan (x)\forall k\in \mathbb{Z}$$

PROOF

TODO

Theorem: Continuity of the Real Tangent Function

The Real Tangent Function is continuous (everywhere it is defined).

PROOF

TODO

Theorem: Derivative of the Real Tangent Function

The Real Tangent Function is differentiable (everywhere it is defined) and its derivative is the recirpocal of the square of the Real Cosine Function:

$$(\tan x{)}^{\prime }=\frac{1}{{\cos }^{2}x}=1+{\tan }^{2}x$$

PROOF

$$(\tan x{)}^{\prime }={\left(\frac{\sin x}{\cos x}\right)}^{\prime }=\frac{(\sin x{)}^{\prime }\cos x-(\cos x{)}^{\prime }\sin x}{{\cos }^{2}x}=\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}$$