Trigonometric Function Domain Range - Understand With Graphs

Before discussing trigonometric function domain range, let's brush up on what a domain and range of any function is. A function is nothing but defined as the rules which are applied to the values inputted. The domain of a function f(x) is defined as the set of all possible values of x such that function f(x) is well defined whereas the range of the function f(x) is defined as the set of all possible values the function f(x) can consider, where x is any number from the domain of a function.  

In other words, the set of all possible values that can be used as an input for the function is the domain of a function. For example, for the function f(k) = \[\sqrt{k}\], the input value can never be negative as the square root of the negative number is not a real number. On the other hand, the range is the set of all output values for the different values that are inputted. For example,  for the function f(x) = x² + 2, the range would be { 2,3,4}.

Domain Range of Trigonometric Function Along With Their Graphs

Mentioned below are the domain and range of all trigonometric functions such as sine, cosine, tan, sec, cosec, and cot along with their graph for better understanding.

Domain and Range For Sine Function 

y = f(x) = sin (x)

Domain: The domain of the sine function is determined for all x real values 

Range : - 1 y 1

Period: 2 = 360º

Here, the sine function is an odd function

Domain And Range For Cosine Function 

y = f(x) = cos(x)

Domain: The domain of cosine function is determined for all x real values 

Range : - 1 y 1

Period: 2 = 360º

Here cosine function is an even function

In the above graph, we can observe that the graph reduces but the range remains the same.

Domain and Range of Tangent Function

Here function (y) = tan (x) comprises vertical asymptotes at \[\pm \frac{(2n+1)\pi }{2}\]. Hence, for - y = f(x) = tan (x).

Domain = The domain of tangent function defined for all x values, except x (2n + 1)(n/2), where n is an integer.

Range = All real numbers ( or y R). 

Period: 2

Here, tangent is an odd function.

Domain and Range For Secant Function

y = f(x) = sec(x)

Domain: The domain of secant function is defined for all real values of x except x ≠(2n + 1)(π/2) where n is any value of the integer

Range: (-∞,-1] ∪ [1,∞)

Period: 2π

Secant is an even function

Domain and Range for Cosecant Function

y = f(x) = cosec(x)

Domain: The domain of the cosecant function is defined for all x real values; except x ≠nπ, where n is an integer value.

Range :(-∞,-1] ∪ [1,∞)

Period: 2

Cosecant is an odd function.

Domain and Range for Cotangent Function

y = f(x) = cot(x)

Domain: The domain of cotangent function for all the x real values; except x ≠nπ, where n is any value of an integer

Range: All the real numbers

Period: π

Cotangent is an odd function

Domain and Range of Trigonometric Function: Tabular Form

To make the students understand the trigonometry domain and range more clearly, we have constructed a table below which clearly describes the domain and range of trigonometric functions.

Domain and Range of Inverse Trigonometric Function

Following are the domain and range of all inverse trigonometric functions such as arcsin, arccosine, arctan, arcsec, arccosec, and arccot along with their graph for better understanding.

ArcSin

The inverse sine function y = \[\sin^{-1}(x)\] means x = sin y.

The inverse sine function is often known as the arcsine function and represented as arcsin x. 

y = \[\sin^{-1}\] has domain [-1, 1] and range \[[\frac{-\pi}{2}, \frac{-\pi}{2}]\].

ArcCosine

The inverse cosine function y = \[\cos^{-1}(x)\] means x = cos y. 

The inverse cosine function is often known as the arccosine function and represented as arccos x. 

y = \[\cos^{-1}\] has domain [-1, 1] and range \[[0,\pi]\].

ArcTan

The inverse tangent function y = \[\tan^{-1}(x)\] means x = tan y. 

The inverse tangent function is also known as the arctangent function and is written as arctangent x. 

ArcTan y = \[\tan^{-1}\] has a restricted domain of all real number values and range \[[\frac{-\pi}{2}, \frac{\pi}{2}]\].

ArcSecant

The inverse secant function, written as arcsecant or \[\sec^{-1}x\] is a function whose domain is R - (1,1)  whereas the range of arcsecant or \[\sec^{-1}x\] is \[[0,\pi]\], -\[{\frac{\pi}{2}}\].

ArcCosecant

The inverse cosecant function, written as arccosecant or \[\csc^{-1}x\] is a function whose domain is R  whereas the range of arccosecant or \[\csc^{-1}x\] is \[[\frac{\pi}{2}, \frac{\pi}{2}]\].

ArcCotangent

The inverse cotangent function, written as arccotangent or \[\cot^{-1}x\] is a function whose domain is R  whereas the range of arccotangent or \[\cot^{-1}x\] is \[[0,\pi]\].

Domain and Range of Inverse Trigonometric Function

To make the students understand the inverse trigonometry domain and range more clearly, we have constructed a table below which clearly describes the domain and range of inverse trig functions.

Solved Example

1. What is the domain and range of y = cos(x) – 5?

Solution:

Here,

Domain: x ∈ R

Range: - 4 ≤ y ≤ - 3, y ∈ R

Notice that the range is simply shifted down by 5 units.

2. What is the value of \[\sin^{-1} \frac{-1}{2}\]?

Solution:

\[\sin^{-1} \frac{-1}{2} = y\]

\[\sin y = \frac{-1}{2} = \sin(\frac{-\pi}{6})\]

y = \[\frac{-\pi}{6}\]

Conclusion

Sin, cos, tan, cot, cosec, and sec are the six trigonometric functions. The angle and the resultant value, respectively, define the domain and range of trigonometric functions. Angles in degrees or radians are the domain of trigonometric functions, and the range is a real integer. Depending on the region where the trigonometric function is not specified, some values are omitted from the domain and range of trigonometric functions. In this post, we looked at the domain and range of trigonometric functions using a table and graph, as well as the domain and range of inverse trigonometric functions using examples.