Notely

9.1 Q-9

Exercise 9.1 — Question 9

Problem

Find the domain and range of the following trigonometric functions, and state whether each is even, odd, or neither, and give its period:

$y = sin x$
$y = cos x$
$y = tan x$
$y = cot x$
$y = sec x$
$y = csc x$

Solution

Domain and Range of Trigonometric Functions

Function	Domain	Range
$sin x$	$R$ (all real numbers)	$[- 1, 1]$
$cos x$	$R$	$[- 1, 1]$
$tan x$	$R ∖ {(2 n + 1) \frac{π}{2} : n \in Z}$	$R$
$cot x$	$R ∖ {nπ : n \in Z}$	$R$
$sec x$	$R ∖ {(2 n + 1) \frac{π}{2} : n \in Z}$	$(- \infty, - 1] \cup [1, + \infty)$
$csc x$	$R ∖ {nπ : n \in Z}$	$(- \infty, - 1] \cup [1, + \infty)$

Even, Odd, and Periodicity

Recall:

A function $f$ is even if $f (- x) = f (x)$ for all $x$ in its domain.
A function $f$ is odd if $f (- x) = - f (x)$ for all $x$ in its domain.
A function $f$ is periodic with period $T$ if $f (x + T) = f (x)$ for all $x$ .

Function	Even / Odd	Period
$sin x$	Odd — $sin (- x) = - sin x$	$2 π$
$cos x$	Even — $cos (- x) = cos x$	$2 π$
$tan x$	Odd — $tan (- x) = - tan x$	$π$
$cot x$	Odd — $cot (- x) = - cot x$	$π$
$sec x$	Even — $sec (- x) = sec x$	$2 π$
$csc x$	Odd — $csc (- x) = - csc x$	$2 π$

Key Points to Remember

$sin x$ and $cos x$ are bounded functions with range $[- 1, 1]$ .
$tan x$ and $cot x$ are unbounded (range = $R$ ) and have the smallest period $π$ .
$sec x$ and $csc x$ are unbounded with a gap between $- 1$ and $1$ in their range.
Even functions are symmetric about the $y$ -axis; odd functions have rotational symmetry about the origin.