9.1 Q-7 | Trigonometric Functions | Mathematics 11

Exercise 9.1 — Question 7

Problem

Find the domain and range of each of the following trigonometric functions, and state whether each function is even, odd, or neither. Also determine the period and find the maximum and minimum values where applicable.

(Refer to your FBISE textbook for the specific sub-parts listed under Q7.)

Key Concepts

Domain and Range of Basic Trigonometric Functions

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

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).

Function	Even / Odd
$cos x$	Even
$sec x$	Even
$sin x$	Odd
$tan x$	Odd
$cot x$	Odd
$csc x$	Odd

Periodicity

A function $f$ is periodic with period $T$ if $f (x + T) = f (x)$ for all $x$ .

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

Maximum and Minimum Values

For a function of the form $f (x) = a sin (b x + c) + d$ or $f (x) = a cos (b x + c) + d$ :

$Maximum value = ∣ a ∣ + d$ $Minimum value = - ∣ a ∣ + d$

Example: For $f (x) = 3 sin x + 2$ :

Maximum $= 3 + 2 = 5$
Minimum $= - 3 + 2 = - 1$

For expressions involving $sin^{2} x$ or $cos^{2} x$ , use the substitution $sin^{2} x \in [0, 1]$ to find the range.

Example: For $f (x) = 2 sin^{2} x - 1$ :

Since $sin^{2} x \in [0, 1]$ : minimum $= 2 (0) - 1 = - 1$ , maximum $= 2 (1) - 1 = 1$

Exercise 9.1 — Question 7

Problem

(Refer to your FBISE textbook for the specific sub-parts listed under Q7.)

Key Concepts

Domain and Range of Basic Trigonometric Functions

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

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).

Function	Even / Odd
$cos x$	Even
$sec x$	Even
$sin x$	Odd
$tan x$	Odd
$cot x$	Odd
$csc x$	Odd

Periodicity

A function $f$ is periodic with period $T$ if $f (x + T) = f (x)$ for all $x$ .

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

Maximum and Minimum Values

For a function of the form $f (x) = a sin (b x + c) + d$ or $f (x) = a cos (b x + c) + d$ :

$Maximum value = ∣ a ∣ + d$ $Minimum value = - ∣ a ∣ + d$

Example: For $f (x) = 3 sin x + 2$ :

Maximum $= 3 + 2 = 5$
Minimum $= - 3 + 2 = - 1$

For expressions involving $sin^{2} x$ or $cos^{2} x$ , use the substitution $sin^{2} x \in [0, 1]$ to find the range.

Example: For $f (x) = 2 sin^{2} x - 1$ :

Since $sin^{2} x \in [0, 1]$ : minimum $= 2 (0) - 1 = - 1$ , maximum $= 2 (1) - 1 = 1$