Mathematical Formulas for Industrial and Mechanical Engineering

Properties of the Tangent Function

1. The tan function is an odd function since tan(2x) 52tan x

2. The tan function is not continuous since tan x 5 sin x=cos x,soy 5 tan x is undefined

when cos x 5 0, that is, when x 56

π

2

, 6

3π

2

, 6

5π

2

, ...

These lines x 56

π

2

, x 56

3π

2

, ... are called vertical asymptotes.

The curve approaches these lines but does not touch them.

3. The tangent function is periodic with period π since tan x 5 tan (x 1 π)

4. tan xAℜ, that is, range 5 ℜ

5. The curve cuts the x-axis at 0, 6 π, 6 2π, ...

Graph of tan x:

y

–ππ

x

0

Graph of y 5

1

tan x

5 cot x :

–π

π

2

0

–

π

π

2

3π

2

y

x

5.20 Sequences and Series

A sequence of numbers is simply a list of numbers,