Domain and Range of Trigonometric Functions
There are six trigonometric functions sin θ, cos θ, tan θ, cot θ, tan θ, cosec θ, and sec θ. The domain and range of trigonometric functions are given by the angle θ and the resultant value, respectively. The domain of the trigonometric functions are angles in degrees or radians and the range is a real number. Some values are excluded from the domain and range of trigonometric functions depending upon the region where the trigonometric function is not defined.
In this article, we will explore the domain and range of trigonometric functions using table, and graph, and domain and range of inverse trigonometric functions along with examples for a better understanding.
What is Domain and Range of Trigonometric Functions?
The domain and range of trigonometric functions are the input values and the output values of trigonometric functions, respectively. The domain of trigonometric functions denotes the values of angles where the trigonometric functions are defined and the range of trigonometric functions gives the resultant value of the trigonometric function corresponding to a particular angle in the domain. There are six main trigonometric functions, namely sin θ, cos θ, tan θ, cot θ, tan θ, cosec θ, and sec θ.
Domain and Range of Trigonometric Function: Sine
We know that sine function is the ratio of the perpendicular and hypotenuse of a rightangled triangle. The domain and range of trigonometric function sine are given by:
 Domain = All real numbers, i.e., (−∞, ∞)
 Range = [1, 1]
Domain and Range of Trigonometric Function: Cosine
We know that the cosine function is the ratio of the adjacent side and hypotenuse of a rightangled triangle. The domain and range of trigonometric function cosine are given by:
 Domain = All real numbers, i.e., (−∞, ∞)
 Range = [1, 1]
Domain and Range of Trigonometric Function: Tangent
We know that the tangent function is the ratio of the opposite and adjacent sides of a rightangled triangle. It can also be written as the ratio of sine and cosine function, therefore the domain of tan x does not contain values where cos x is equal to zero. We know that cos x is 0 at odd integral multiples of π/2, hence the domain and range of trigonometric function tangent are given by:
 Domain = R  (2n + 1)π/2
 Range = (−∞, ∞)
Domain and Range of Trigonometric Function: Cotangent
We know that the cotangent function is the ratio of the adjacent side and the opposite side in a rightangled triangle. It can also be written as the ratio of cosine and sine function, and cot x is the reciprocal of tan x. Therefore the domain of cot x does not contain values where sin x is equal to zero. We know that sin x is 0 at integral multiples of π, hence the domain and range of trigonometric function cotangent are given by:
 Domain = R  nπ
 Range = (−∞, ∞)
Domain and Range of Trigonometric Function: Secant
We know that the secant function is the ratio of the hypotenuse and the adjacent side in a rightangled triangle. It can also be written as the reciprocal of the cosine function. Therefore the domain of sec x does not contain values where cos x is equal to zero. We know that cos x is 0 at odd integral multiples of π, hence the domain and range of trigonometric function secant are given by:
 Domain = R  (2n + 1)π/2
 Range = (∞, 1] U [+1, +∞)
Domain and Range of Trigonometric Function: Cosecant
We know that the cosecant function is the ratio of the hypotenuse and the opposite side in a rightangled triangle. It can also be written as the reciprocal of the sine function. Therefore the domain of trigonometric function cosec x does not contain values where sin x is equal to zero. We know that sin x is 0 at integral multiples of π, hence the domain and range of trigonometric function cosecant are given by:
 Domain = R  nπ
 Range = (∞, 1] U [+1, +∞)
Domain and Range of Trigonometric Functions Table
Now, we have studied the domain and range of trigonometric functions. The below table gives the summary of it which will help for a better understanding and using for solving various problems:
Trigonometric Functions  Domain  Range 
Sinθ  (∞, + ∞)  [1, +1] 
Cosθ  (∞ +∞)  [1, +1] 
Tanθ  R  (2n + 1)π/2  (∞, +∞) 
Cotθ  R  nπ  (∞, +∞) 
Secθ  R  (2n + 1)π/2  (∞, 1] U [+1, +∞) 
Cosecθ  R  nπ  (∞, 1] U [+1, +∞) 
Domain and Range of Inverse Trigonometric Functions
A function is invertible if and only if it is bijective. The inverse trigonometric functions are the inverse of the trigonometric functions and to make the trigonometric functions invertible, we restrict their domains to the principal value branch. The table below represents the domain and range of the inverse trigonometric functions:
Inverse Trigonometric Functions  Domain  Range 
Sin^{1}x  [1, +1]  [π/2, π/2] 
Cos^{1}x  [1, +1]  [0, π] 
Tan^{1}x  (∞, + ∞)  (π/2, π/2) 
Cot^{1}x  (∞, + ∞)  (0, π) 
Sec^{1}x  (−∞,−1] U [1,∞)  [0, π/2) U (π/2, π] 
Cosec^{1}x  (−∞,−1] U [1,∞)  [π/2, 0) U (0, π/2] 
Domain and Range of Trigonometric Functions Using Graph
Next, we will explore the domain and range of trigonometric functions using graphs of the trigonometric functions. Given below are the graphs of the six trigonometric functions. As we can see in the graphs, the domain and range of the trigonometric functions are represented by the xaxis and yaxis, respectively.
Tips and Tricks on Domain and Range of Trigonometric Functions
 Check for the value of input where the function is not defined. The value where the function is not defined can be excluded from the domain.
 The range of a trigonometric function is given by the output values for each of the input values (domain).
Related Topics
Domain and Range of Trigonometric Functions Examples

Example 1: Find the domain and range of y = 3 tan x.
Solution: We know that the domain and range of trigonometric function tan x is given by, Domain = R  (2n + 1)π/2, Range = (∞, +∞)
Note that the domain is given by the values that x can take, therefore the domains of tan x and 3 tan x are the same. Hence the domain of y = 3 tan x is R  (2n + 1)π/2
The range of tan x is (∞, +∞) ⇒ ∞ < y < ∞
⇒ ∞ < tan x < ∞
⇒ ∞ < 3 tan x < ∞ [As multiplication of ∞ by 3 results in ∞ only]
Therefore, the range of y = 3 tan x is (∞, ∞).
Answer: Domain = R  (2n + 1)π/2, Range = (∞, ∞)

Example 2: Determine the domain and range of y = sin x  3
Solution: We know that the domain and range of sin x are (∞, + ∞) and [1, 1], respectively.
As sin x is defined for all real numbers and y = sin x  3 is defined for all real numbers, therefore the domain for y = sin x  3 is (∞, + ∞).
Now, to determine the range, we need to determine the interval for y.
We have 1 ≤ sin x ≤ 1 ⇒ 1  3 ≤ sin x  3 ≤ 1  3 ⇒ 4 ≤ sin x  3 ≤ 2 ⇒ 4 ≤ y ≤ 2. Therefore, the range of y = sin x  3 is [4, 2].
Answer: Domain = (∞, + ∞), Range = [4, 2]
FAQs on Domain and Range of Trigonometric Functions
What is the Domain and Range of Trigonometric Functions in Trigonometry?
The domain and range of trigonometric functions are the input values and the output values of trigonometric functions, respectively.
 For sin θ, Domain = (∞, + ∞), Range = [1, 1]
 For cos θ, Domain = (∞, + ∞), Range = [1, 1]
 For tan θ, Domain = R  (2n + 1)π/2, Range = (∞, +∞)
 For cot θ, Domain = R  nπ , Range = (∞, +∞)
 For sec θ, Domain = R  (2n + 1)π/2, Range = (∞, 1] U [+1, +∞)
 For cosec θ, Domain = R  nπ , Range = (∞, 1] U [+1, +∞)
How To Find Domain and Range of Trigonometric Functions?
Domain and Range of Trigonometric Functions can be found by checking where the function is defined and the output values of the function for each input value.
What is the Range of Cos Square theta?
We know that the range of cos θ is [1, 1], and cos^{2}θ is always positive, therefore the range of cos square theta is [0, 1].
How To Find Domain and Range of Inverse Trigonometric Functions?
The inverse trigonometric functions are the inverse of the trigonometric functions and to make the trigonometric functions invertible, we restrict their domains to the principle value branch.
What is Domain and Range of Sec Theta?
The domain of sec θ is R  (2n + 1)π/2 and the range is (∞, 1] U [+1, +∞), where n is an integer.
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