Trigonometric

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facets

Rational functions were quotients of polynomials. We can do the same with trigonometric functions.

Tangent

Tangent is the quotient of sine and cosine

$\tan (\theta ) = \frac {\sin (\theta )}{\cos (\theta )}$

Its domain is all real numbers except the zeros of cosine, $\left \{ \frac {(2k+1)\pi }{2} \, | \, k \in \textbf {Z} \right \}$ .

At each of these singularities, the graph has a vertical asymptote, where the function becomes unbounded.

Graph of $z = \tan (\theta )$ .

Tangent takes on every value in $\mathbb {R}$ exactly once inside the open interval $\left ( \frac {-\pi }{2}, \frac {\pi }{2} \right )$ . It is then periodic with a period of $\pi$ .

Tangent is always increasing decreasing inside its domain.

Tangent is unbounded and has no global or local maximums or minimums.

Tangent has an infinite number of zeros. They coincide with the zeros of : $\{ k \pi \, | \, k \in \textbf {Z} \}$

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