**Domain and Range of Trigonometric functions : **

In this section you will learn how to find domain and range of a trigonometric function.

To make the students to understand domain and range of a trigonometric function, we have given a table which clearly says the domain and range of trigonometric functions.

In any right angle triangle, we can define the following six trigonometric ratios.

sin x, cos x, csc x, sec x, tan x, cot x

In the above six trigonometric ratios, the first two trigonometric ratios sin x and cos x are defined for all real values of x.

The two trigonometric ratios sin x and cos x are defined for all real values of x.

So, the domain for sin x and cos x is all real numbers.

The diagrams given below clearly explains the range of sin x and cos x.

**Range of sin x**

**Range of cos x**

From the pictures above, it is very clear that the range of y = sin x and y = cos x is

{y | -1 ≤ y ≤ 1}

We know that sin (k∏) = 0, cos [(2k+1)∏] /2 = 0, here "k" is an integer.

Then,

k = ...........-2, -1, 0, 1, 2, ..........

For k = -2,

sin (-2∏) = 0 and cos (-3∏/2) = 0

For k = -1

sin (-∏) = 0 and cos (-∏/2) = 0

For k = 0,

sin (0) = 0 and cos (∏/2) = 0

For k = 1,

sin (∏) = 0 and cos (3∏/2) = 0

For k = 2,

sin (2∏) = 0 and cos (5∏/2) = 0

**Stuff 1 :**

We know that csc x and sec x are the reciprocals of sin x and cos x respectively.

Let us see the values of csc x for

x = .......-2∏, -∏, 0, ∏, 2∏, .........

csc(-2∏) = 1/sin(-2∏) = 1/0 = Undefined

csc(-∏) = 1/sin(-∏) = 1/0 = Undefined

csc(0) = 1/sin(0) = 1/0 = Undefined

csc(∏) = 1/sin(∏) = 1/0 = Undefined

csc(2∏) = 1/sin(2∏) = 1/0 = Undefined

From the above examples, it is very clear, that csc x is defined for all real values of x except

x = .......-2∏, -∏, 0, ∏, 2∏, .........

So the domain of csc x is

{x | x ≠ ...-2∏, -∏, 0, ∏, 2∏, ..}

In the same way, domain of sec x is

{x | x≠ ...-3∏/2, -∏/2, ∏/2, 3∏/2, 5∏/2 ...}

Let y = csc x.

In the trigonometric function y = csc x, when plug values for x such that

x ∈ R - {....-2∏, -∏, 0, ∏, 2∏,....},

we will get values for "y" which are out of the interval

(-1, 1)

So the range of csc x is

{y | y ≤ -1 or y ≥ 1}

In the same way, for the function y = sec x, when plug values for x such that

x ∈ R - {...-3∏/2, -∏/2, ∏/2, 3∏/2, 5∏/2 ..},

we will get values for y which are out of the interval

(-1, 1)

So the range of sec(x) is

{ y | y ≤ -1 or y ≥ 1}

The trigonometric function tan x will become undefined for

x = [(2k + 1)∏] / 2

here k is an integer.

Substituting k = ...........-2, -1, 0, 1, 2, .......... we get

x = ..........-3∏/2, -∏/2, ∏/2, 3∏/2, 5∏/2........

For the above values of x, tan x becomes undefined and tan x is defined for all other real values.

Therefore, domain of tan x is

{x | x≠..-3∏/2,-∏/2,∏/2,3∏/2,5∏/2...}

The trigonometric function cot x will become undefined for

x = k∏

here k" is an integer.

Substituting k = ...........-2, -1, 0, 1, 2, .......... we get

x = ..........-2∏, -∏, 0, ∏, 2∏.......

For the above values of x, cot x becomes undefined and cot(x) is defined for all other real values.

So, the domain of cot x is

{x | x≠..-2∏, -∏, 0, ∏, 2∏...}

In the trigonometric function y = tan x, if we substitute values for x such that

x ∈ R - {...-3∏/2, -∏/2, ∏/2, 3∏/2, 5∏/2 ..},

we will get all real values for "y" .

So the range of tan x is

All Real Values

In the same way, for cot x, if we substitute values for x such that

x ∈ R - {...-2∏, -∏, 0, ∏, 2∏ ..},

we will get all values for y.

So the range of cot x is

All Real Values

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