9.1 Maximum and Minimum Values of Trigonometric Functions | Trigonometric Functions | Mathematics 11

Domain and Range of Trigonometric Functions

The six trigonometric functions have the following domains and ranges:

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

Even and Odd Trigonometric Functions

A function $f (x)$ is:

Even if $f (- x) = f (x)$ for all $x$ in its domain (graph symmetric about the y-axis).
Odd if $f (- x) = - f (x)$ for all $x$ in its domain (graph symmetric about the origin).

Classification of Trig Functions

Even	Odd
$cos (- x) = cos x$	$sin (- x) = - sin x$
$sec (- x) = sec x$	$tan (- x) = - tan x$
	$cot (- x) = - cot x$
	$csc (- x) = - csc x$

Example: Show that $f (x) = \frac{x ^{2} tan x}{x + sin x}$ is even.

$f (- x) = \frac{( - x ) ^{2} t a n ( - x )}{( - x ) + s i n ( - x )} = \frac{x ^{2} ( - t a n x )}{- ( x + s i n x )} = \frac{- x ^{2} t a n x}{- ( x + s i n x )} = \frac{x ^{2} t a n x}{x + s i n x} = f (x)$

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

Periodicity of Trigonometric Functions

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

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

Period of Transformed Functions

$Period of sin (a x) or cos (a x) = \frac{2 π}{∣ a ∣}$

$Period of tan (a x) or cot (a x) = \frac{π}{∣ a ∣}$

Example: Find the period of $y = sin (\frac{a x}{b})$ .

Here $k = \frac{a}{b}$ , so period $= \frac{2 π}{a / b} = \frac{2 π b}{a}$ .

Period of a Sum of Two Trig Functions

For $y = f (x) + g (x)$ , find the period of each term separately, then take their LCM.

Example: Find the period of $y = sin x + cos 2 x$ .

Period of $sin x = 2 π$
Period of $cos 2 x = π$
LCM $(2 π, π) = 2 π$

So the period of $y$ is $2 π$ .

Maximum and Minimum Values

Standard Form: $y = a + b sin (c x + d)$ or $y = a + b cos (c x + d)$

Since $- 1 \leq sin (c x + d) \leq 1$ :

$Maximum = a + ∣ b ∣, Minimum = a - ∣ b ∣$

Example: Find the max and min of $y = 5 - 3 cos (2 θ)$ .

Here $a = 5$ , $b = - 3$ , so $∣ b ∣ = 3$ .

Maximum $= 5 + 3 = 8$ (when $cos 2 θ = - 1$ )
Minimum $= 5 - 3 = 2$ (when $cos 2 θ = 1$ )

Reciprocal Functions: $y = \frac{1}{f ( x )}$ where $f (x) > 0$

When the denominator is smallest, $y$ is largest, and vice versa:

$Maximum of y = \frac{1}{m i n f ( x )}, Minimum of y = \frac{1}{m a x f ( x )}$

Example: Find the max and min of $y = \frac{1}{3 + 2 sin θ}$ .

$max (3 + 2 sin θ) = 3 + 2 = 5$ (when $sin θ = 1$ )
$min (3 + 2 sin θ) = 3 - 2 = 1$ (when $sin θ = - 1$ )
Maximum of $y = \frac{1}{1} = 1$
Minimum of $y = \frac{1}{5}$

Graphs and Properties of $sin θ$ , $cos θ$ , and $tan θ$

$y = sin θ$

Domain: $R$ | Range: $[- 1, 1]$
Period: $2 π$ | Amplitude: $1$
Parity: Odd — symmetric about the origin
Passes through $(0, 0)$ ; maximum at $θ = π /2$ ; minimum at $θ = 3 π /2$

$y = cos θ$

Domain: $R$ | Range: $[- 1, 1]$
Period: $2 π$ | Amplitude: $1$
Parity: Even — symmetric about the y-axis
Maximum at $θ = 0$ ; zero crossings at $θ = \pm π /2$

$y = tan θ$

Domain: $R ∖ {\frac{( 2 n + 1 ) π}{2}}$ | Range: $(- \infty, \infty)$
Period: $π$ | No amplitude (unbounded)
Parity: Odd — symmetric about the origin
Vertical asymptotes at $θ = \pm \frac{π}{2}, \pm \frac{3 π}{2}, \dots$

Graphical Solution of Equations

To solve $cos x = x$ graphically: plot $y = cos x$ and $y = x$ on the same axes. The x-coordinate(s) of their intersection point(s) are the solutions.

Domain and Range of Trigonometric Functions

The six trigonometric functions have the following domains and ranges:

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

Even and Odd Trigonometric Functions

A function $f (x)$ is:

Even if $f (- x) = f (x)$ for all $x$ in its domain (graph symmetric about the y-axis).
Odd if $f (- x) = - f (x)$ for all $x$ in its domain (graph symmetric about the origin).

Classification of Trig Functions

Even	Odd
$cos (- x) = cos x$	$sin (- x) = - sin x$
$sec (- x) = sec x$	$tan (- x) = - tan x$
	$cot (- x) = - cot x$
	$csc (- x) = - csc x$

Example: Show that $f (x) = \frac{x ^{2} tan x}{x + sin x}$ is even.

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

Periodicity of Trigonometric Functions

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

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

Period of Transformed Functions

$Period of sin (a x) or cos (a x) = \frac{2 π}{∣ a ∣}$

$Period of tan (a x) or cot (a x) = \frac{π}{∣ a ∣}$

Example: Find the period of $y = sin (\frac{a x}{b})$ .

Here $k = \frac{a}{b}$ , so period $= \frac{2 π}{a / b} = \frac{2 π b}{a}$ .

Period of a Sum of Two Trig Functions

For $y = f (x) + g (x)$ , find the period of each term separately, then take their LCM.

Example: Find the period of $y = sin x + cos 2 x$ .

Period of $sin x = 2 π$
Period of $cos 2 x = π$
LCM $(2 π, π) = 2 π$

So the period of $y$ is $2 π$ .

Maximum and Minimum Values

Standard Form: $y = a + b sin (c x + d)$ or $y = a + b cos (c x + d)$

Since $- 1 \leq sin (c x + d) \leq 1$ :

$Maximum = a + ∣ b ∣, Minimum = a - ∣ b ∣$

Example: Find the max and min of $y = 5 - 3 cos (2 θ)$ .

Here $a = 5$ , $b = - 3$ , so $∣ b ∣ = 3$ .

Maximum $= 5 + 3 = 8$ (when $cos 2 θ = - 1$ )
Minimum $= 5 - 3 = 2$ (when $cos 2 θ = 1$ )

Reciprocal Functions: $y = \frac{1}{f ( x )}$ where $f (x) > 0$

When the denominator is smallest, $y$ is largest, and vice versa:

$Maximum of y = \frac{1}{m i n f ( x )}, Minimum of y = \frac{1}{m a x f ( x )}$

Example: Find the max and min of $y = \frac{1}{3 + 2 sin θ}$ .

$max (3 + 2 sin θ) = 3 + 2 = 5$ (when $sin θ = 1$ )
$min (3 + 2 sin θ) = 3 - 2 = 1$ (when $sin θ = - 1$ )
Maximum of $y = \frac{1}{1} = 1$
Minimum of $y = \frac{1}{5}$

Graphs and Properties of $sin θ$ , $cos θ$ , and $tan θ$

$y = sin θ$

Domain: $R$ | Range: $[- 1, 1]$
Period: $2 π$ | Amplitude: $1$
Parity: Odd — symmetric about the origin
Passes through $(0, 0)$ ; maximum at $θ = π /2$ ; minimum at $θ = 3 π /2$

$y = cos θ$

Domain: $R$ | Range: $[- 1, 1]$
Period: $2 π$ | Amplitude: $1$
Parity: Even — symmetric about the y-axis
Maximum at $θ = 0$ ; zero crossings at $θ = \pm π /2$

$y = tan θ$

Domain: $R ∖ {\frac{( 2 n + 1 ) π}{2}}$ | Range: $(- \infty, \infty)$
Period: $π$ | No amplitude (unbounded)
Parity: Odd — symmetric about the origin
Vertical asymptotes at $θ = \pm \frac{π}{2}, \pm \frac{3 π}{2}, \dots$

Graphical Solution of Equations

To solve $cos x = x$ graphically: plot $y = cos x$ and $y = x$ on the same axes. The x-coordinate(s) of their intersection point(s) are the solutions.