Domain And Range Of Tangent

Domain and Range of Trigonometric Functions

At that place 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 bending 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 part is non defined.

In this article, nosotros volition explore the domain and range of trigonometric functions using table, and graph, and domain and range of changed trigonometric functions along with examples for a improve understanding.

1.	What is Domain and Range of Trigonometric Functions?
2.	Domain and Range of Trigonometric Functions Table
3.	Domain and Range of Changed Trigonometric Functions
four.	Domain and Range of Trigonometric Functions Using Graph
5.	FAQs on Domain and Range of Trigonometric Functions

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 bending in the domain. There are half dozen principal 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 part 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 Trigonometric Role: Cosine

Nosotros know that the cosine role is the ratio of the adjacent side and hypotenuse of a right-angled triangle. The domain and range of trigonometric function cosine are given by:

Domain = All real numbers, i.eastward., (−∞, ∞)
Range = [-1, 1]

Domain and Range of Trigonometric Part: Tangent

We know that the tangent function is the ratio of the reverse and side by side sides of a right-angled triangle. It tin can also be written equally 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 10 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/ii
Range = (−∞, ∞)

Domain and Range of Trigonometric Function: Cotangent

Nosotros know that the cotangent office is the ratio of the side by side side and the reverse side in a right-angled triangle. Information technology tin likewise 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 incorporate 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 Part: Secant

We know that the secant function is the ratio of the hypotenuse and the side by side side in a right-angled triangle. Information technology can too be written equally the reciprocal of the cosine function. Therefore the domain of sec x does not incorporate values where cos x is equal to cypher. We know that cos 10 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 role is the ratio of the hypotenuse and the opposite side in a right-angled triangle. Information technology tin can also be written as the reciprocal of the sine role. Therefore the domain of trigonometric function cosec x does not incorporate values where sin x is equal to zero. We know that sin 10 is 0 at integral multiples of π, hence the domain and range of trigonometric function cosecant are given by:

Domain = R - nπ
Range = (-∞, -ane] U [+1, +∞)

Domain and Range of Trigonometric Functions Table

Now, we have studied the domain and range of trigonometric functions. The beneath table gives the summary of it which volition help for a better understanding and using for solving diverse issues:

Trigonometric Functions	Domain	Range
Sinθ	(-∞, + ∞)	[-1, +ane]
Cosθ	(-∞ +∞)	[-i, +1]
Tanθ	R - (2n + 1)π/2	(-∞, +∞)
Cotθ	R - nπ	(-∞, +∞)
Secθ	R - (2n + 1)π/2	(-∞, -i] U [+1, +∞)
Cosecθ	R - nπ	(-∞, -ane] 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 tabular array below represents the domain and range of the inverse trigonometric functions:

Inverse Trigonometric Functions	Domain	Range
Sin^-1ten	[-1, +1]	[-π/two, π/2]
Cos^-1x	[-1, +1]	[0, π]
Tan^-anex	(-∞, + ∞)	(-π/ii, π/2)
Cot^-anex	(-∞, + ∞)	(0, π)
Sec^-aneten	(−∞,−one] U [1,∞)	[0, π/2) U (π/2, π]
Cosec^-110	(−∞,−one] U [ane,∞)	[-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. Every bit we tin can see in the graphs, the domain and range of the trigonometric functions are represented past the ten-axis and y-centrality, respectively.

Domain and Range of Trigonometric Functions

Tips and Tricks on Domain and Range of Trigonometric Functions

Bank check for the value of input where the part is not divers. The value where the function is not divers can be excluded from the domain.
The range of a trigonometric function is given past the output values for each of the input values (domain).

Related Topics

Trigonometry
Trigonometry Formulas
Trigonometric Identities

Domain and Range of Trigonometric Functions Examples

Case 1: Observe the domain and range of y = 3 tan 10.

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 10 can take, therefore the domains of tan x and 3 tan 10 are the same. Hence the domain of y = iii tan x is R - (2n + 1)π/ii

The range of tan 10 is (-∞, +∞) ⇒ -∞ < y < ∞

⇒ -∞ < tan 10 < ∞

⇒ -∞ < 3 tan x < ∞ [As multiplication of ∞ by iii results in ∞ only]

Therefore, the range of y = 3 tan x is (-∞, ∞).

Answer: Domain = R - (2n + one)π/2, Range = (-∞, ∞)
Example 2: Determine the domain and range of y = sin 10 - three

Solution: We know that the domain and range of sin x are (-∞, + ∞) and [-i, i], respectively.

Equally sin ten is defined for all real numbers and y = sin x - 3 is divers for all real numbers, therefore the domain for y = sin x - three is (-∞, + ∞).

Now, to make up one's mind the range, we need to make up one's mind the interval for y.

We have -1 ≤ sin 10 ≤ 1 ⇒ -i - iii ≤ sin x - 3 ≤ 1 - 3 ⇒ -4 ≤ sin x - 3 ≤ -two ⇒ -4 ≤ y ≤ -2. Therefore, the range of y = sin ten - 3 is [-4, -2].

Answer: Domain = (-∞, + ∞), Range = [-4, -2]

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Domain and Range of Trigonometric Functions Questions

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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, ane]
For tan θ, Domain = R - (2n + 1)π/two, Range = (-∞, +∞)
For cot θ, Domain = R - nπ , Range = (-∞, +∞)
For sec θ, Domain = R - (2n + ane)π/ii, Range = (-∞, -1] U [+one, +∞)
For cosec θ, Domain = R - nπ , Range = (-∞, -ane] U [+i, +∞)

How To Detect Domain and Range of Trigonometric Functions?

Domain and Range of Trigonometric Functions can be establish by checking where the role is divers and the output values of the function for each input value.

What is the Range of Cos Foursquare theta?

We know that the range of cos θ is [-1, 1], and cos²θ is always positive, therefore the range of cos foursquare theta is [0, one].

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.