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 is the set of angles in degrees or radians and
- the range is the set of all or some real numbers.
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 are the 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 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 Sine
We know that sine function is the ratio of the perpendicular and hypotenuse of a right-angled 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 Cosine
We know that the cosine function is the ratio of the adjacent side and hypotenuse of a right-angled triangle. The domain and range of cosine are given by:
- Domain = All real numbers, i.e., (−∞, ∞)
- Range = [-1, 1]
☛Note: The domain and range of sin and cos graphs are the same.
Domain and Range of Tangent
We know that the tangent function is the ratio of the opposite and adjacent sides of a right-angled triangle. It can also be written as the ratio of sine and cosine functions, therefore the domain of tan x does not contain values where cos x is equal to zero. We know that cos x = 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 = (−∞, ∞)
Here, 'n' is an integer.
Domain and Range of Cotangent
We know that the cotangent function is the ratio of the adjacent side and the opposite side in a right-angled triangle. It can also be written as the ratio of cosine and sine functions, 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 = 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 Secant
We know that the secant function is the ratio of the hypotenuse and the adjacent side in a right-angled 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 secant are given by:
- Domain = R - (2n + 1)π/2
- Range = (-∞, -1] U [1, ∞)
Since the range of cos x is [-1, 1], the output values of cos x are either proper fractions, -1, or 1, (and of course 0 as well). Thus, when we reciprocate these, we either get improper fractions, -1, or 1 respectively. Accordingly, the range of sec x is defined above.
Domain and Range of Cosecant
We know that the cosecant function is the ratio of the hypotenuse and the opposite side in a right-angled 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 cosecant are given by:
- Domain = R - nπ
- Range = (-∞, -1] U [1, ∞)
The range of cosecant can be derived exactly the same way how we derived for secant.
Domain and Range of Trigonometric Functions Table
Now, we have studied the domain and range of trigonometric functions. The below table gives a summary of it which will help for a better understanding and using for solving various problems:
| Trigonometric Functions | Domain | Range |
|---|---|---|
| sin θ | (-∞, ∞) | [-1, 1] |
| cos θ | (-∞, ∞) | [-1, 1] |
| tan θ | R - (2n + 1)π/2 | (-∞, ∞) |
| cot θ | R - nπ | (-∞, ∞) |
| sec θ | R - (2n + 1)π/2 | (-∞, -1] U [1, ∞) |
| cosec θ | 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-1x | [-1, 1] | [-π/2, π/2] |
| cos-1x | [-1, 1] | [0, π] |
| tan-1x | (-∞, ∞) | (-π/2, π/2) |
| cot-1x | (-∞, ∞) | (0, π) |
| sec-1x | (−∞,−1] U [1,∞) | [0, π/2) U (π/2, π] |
| cosec-1x | (−∞,−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 x-axis and y-axis, 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).
- Also, use the reciprocal identities csc x = 1/sin x, sec x = 1/cos x, and also the identities tan x = sin x/cos x and cot x = cos x/sin x to find the domain and range.
☛Related Topics:
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 what are the output values of the function for each input value.
What are the Domain and Range of Inverse Trig Functions?
The domain and range of inverse trigonometric functions are the set of input values that they take and the set of outputs that they produce respectively. Click here to know the domain and range in detail.
What is the Range of Cos Square theta?
We know that the range of cos θ is [-1, 1], and cos2θ 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. Then we just interchange the domain and range of trig functions to get the domain and range of inverse trig functions.
What is the 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.