9.1 Q-3 | Trigonometric Functions | Mathematics 11

Exercise 9.1 — Question 3

Domain and Range of Trigonometric Functions

The six trigonometric functions have specific domains (where they are defined) and ranges (the set of output values).

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

Key point: $tan x$ and $sec x$ are undefined where $cos x = 0$ , i.e., at odd multiples of $\frac{π}{2}$ . Similarly, $cot x$ and $csc x$ are undefined where $sin x = 0$ , i.e., at integer multiples of $π$ .

Even and Odd Trigonometric Functions

A function $f$ is even if $f (- x) = f (x)$ for all $x$ in its domain (symmetric about the $y$ -axis).

A function $f$ is odd if $f (- x) = - f (x)$ for all $x$ in its domain (symmetric about the origin).

Even functions: $cos (- x) = cos x sec (- x) = sec x$

Odd functions: $sin (- x) = - sin x tan (- x) = - tan x$ $cot (- x) = - cot x csc (- x) = - csc x$

Periodicity of Trigonometric Functions

A function $f$ is periodic with period $T$ if $f (x + T) = f (x)$ for all $x$ in its domain, and $T$ is the smallest such positive number.

Function	Period
$sin x$	$2 π$
$cos x$	$2 π$
$tan x$	$π$
$cot x$	$π$
$sec x$	$2 π$
$csc x$	$2 π$

Verification example: $sin (x + 2 π) = sin x \cdot cos 2 π + cos x \cdot sin 2 π = sin x \cdot 1 + cos x \cdot 0 = sin x ✓$

Worked Example

Q: State whether $f (x) = sin x \cdot cos x$ is even, odd, or neither. Also state its period.

Solution:

Even/Odd check: $f (- x) = sin (- x) cos (- x) = (- sin x) (cos x) = - sin x cos x = - f (x)$

Since $f (- x) = - f (x)$ , the function is odd.

Period: Using the identity $sin x cos x = \frac{1}{2} sin 2 x$ , the period is $\frac{2 π}{2} = π$ .

Exercise 9.1 — Question 3

Domain and Range of Trigonometric Functions

The six trigonometric functions have specific domains (where they are defined) and ranges (the set of output values).

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

Even and Odd Trigonometric Functions

A function $f$ is even if $f (- x) = f (x)$ for all $x$ in its domain (symmetric about the $y$ -axis).

A function $f$ is odd if $f (- x) = - f (x)$ for all $x$ in its domain (symmetric about the origin).

Even functions: $cos (- x) = cos x sec (- x) = sec x$

Odd functions: $sin (- x) = - sin x tan (- x) = - tan x$ $cot (- x) = - cot x csc (- x) = - csc x$

Periodicity of Trigonometric Functions

A function $f$ is periodic with period $T$ if $f (x + T) = f (x)$ for all $x$ in its domain, and $T$ is the smallest such positive number.

Function	Period
$sin x$	$2 π$
$cos x$	$2 π$
$tan x$	$π$
$cot x$	$π$
$sec x$	$2 π$
$csc x$	$2 π$

Verification example: $sin (x + 2 π) = sin x \cdot cos 2 π + cos x \cdot sin 2 π = sin x \cdot 1 + cos x \cdot 0 = sin x ✓$

Worked Example

Q: State whether $f (x) = sin x \cdot cos x$ is even, odd, or neither. Also state its period.

Solution:

Even/Odd check: $f (- x) = sin (- x) cos (- x) = (- sin x) (cos x) = - sin x cos x = - f (x)$

Since $f (- x) = - f (x)$ , the function is odd.

Period: Using the identity $sin x cos x = \frac{1}{2} sin 2 x$ , the period is $\frac{2 π}{2} = π$ .