DOMAIN AND RANGE OF TRIGONOMETRIC FUNCTIONS

About the topic "Domain and range of trigonometric functions"

"Domain and range of trigonometric functions" is a much needed stuff required by almost all the students who study math in high schools.

Even though students can get this stuff on internet, they do not understand exactly what has been explained. 

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

Domain of sin(x) and cos(x)

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.

In the topic, "Domain and Range of trigonometric functions", next we are going to see the range of sin(x) and cos(x).

Range of sin(x) and cos(x)

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}

In the topic, "Domain and Range of trigonometric functions", next we are going to see the domain of csc(x) and sec(x).

Domain of csc(x) and sec(x)

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

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

we get, 

      sin (-2∏) = 0                 cos (-3∏/2) = 0                     for k = -2

      sin (-∏) = 0                   cos (-∏/2) = 0                       for k = -1

      sin (0) = 0                     cos (∏/2) = 0                        for k = 0

      sin (∏) = 0                    cos (3∏/2) = 0                      for k = 1

      sin (2∏) = 0                  cos (5∏/2) = 0                     for k = 2

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) =  { x | x ≠ ...-2, -, 0, , 2, ..}

In the same way,

Domain of sec(x) ={x|x≠ ...-3∏/2, -∏/2, ∏/2, 3∏/2, 5∏/2 ...}

In the topic, "Domain and Range of trigonometric functions", next we are going to see the range of csc(x) and sec(x).

Range of csc(x) and sec(x)

In the trigonometric function 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) = { y | y≤-1 or y≥1}

In the same way, for 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 csc(x) = { y | y≤-1 or y≥1}

In the topic, "Domain and Range of trigonometric functions", next we are going to see the domain of tan(x) and cot(x).

Domain of tan(x) and cot(x)

The trigonometric function tan(x) will become undefined for

 x = [(2k+1)∏] /2 , here "k" is an integer.

if take 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) ={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.

if take 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) ={x|x≠..-2∏, -∏, 0, ∏, 2∏...}

In the topic, "Domain and Range of trigonometric functions", next we are going to see the range of tan(x) and cot(x).

Range of tan(x) and cot(x)

In the trigonometric function tan(x), when plug 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) = All Real Values

In the same way, for cot(x), when plug values for "x" such that         x∈R - {...-2∏, -∏, 0, ∏, 2∏ ..}, we will get all values for "y". 

So the range of cot(x) = All Real Values

You can also visit the following websites to know more about "Domain and Range of Trigonometric Functions"

http://hotmath.com

http://www.analyzemath.com

https://www.khanacademy.org

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