Domain and Range of Trigonometric Functions

Domain and Range of Trigonometric Functions are the values in which the trigonometric function is defined and the output for the domain respectively. Trigonometric Ratios describe relationships between angles and the sides- of right triangles. A Function defined in terms of Trigonometric Ratios is called a Trigonometric Function.

There are a total of six Trigonometric Functions. They play a significant role in various branches of mathematics and science, particularly in calculus and geometry. Trigonometric functions are essential in mathematics and have widespread applications in various fields.

In this article, we will discuss trigonometric functions in detail and learn more about the domain and range of various trigonometric functions.

What are Trigonometric Functions?

Trigonometric Functions relate the angles and sides of a right triangle. Trigonometric functions, such as sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec), are mathematical functions that relate angles to the ratios of the sides of a right triangle. These functions are defined for all real numbers and have specific domains and ranges. Sine gives the ratio of the opposite side to the hypotenuse, cosine gives the ratio of the adjacent side to the hypotenuse and tangent provides the ratio of the opposite side to the adjacent side.

What is Domain and Range?

Domain of a function is the set of input values (angles, in this case) for which the function is defined and produces a valid output. Trigonometric functions, such as sine, cosine, and tangent, have a domain that includes all real numbers. However, their range varies. Range of a function is the set of all possible output values. To calculate trigonometric values, you use the ratios. For example, sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent.

Example:

Common trigonometric values include sin(30°) = 1/2, cos(45°) = √2/2, and tan(60°) = √3.

Sine and cosine functions have a range of [-1, 1], representing the amplitude of oscillation.

Tangent has a domain restriction excluding odd multiples of π/2 (-π/2, π/2, 3π/2, etc.), but its range is all real numbers.

Read More: Domain, and Range

What are the Domain and Range of Trigonometric Functions?

As we know domain and range of a function tell us, what values we can take and what is the range of the functions. The domain and range of the trigonometric functions are added below,

Domain and Range of Sine Function {Sin(θ)}

The sine function sin(θ) has a domain of all real numbers (−∞, ∞) and a range between -1 and 1 inclusive: -1 ≤ sin(θ) ≤ 1.

Domain and Range of Cosine Function {Cos (θ)}

The cosine function cos(θ) also has a domain of all real numbers and a range between -1 and 1: -1 ≤ cos(θ) ≤ 1.

Domain and Range of Tangent Function {Tan (θ)}

The tangent function, tan(θ) has a domain of all real numbers except the values where the cosine is 0. So, its domain is (-∞, ∞) excluding the values where cos(θ) = 0 which are odd multiples of π/2. Its range is all real numbers covering the entire real number line.

Domain and Range of Cotagent Function {Cot (θ)}

The cotangent function cot(θ) shares its domain with the tangent function excluding the points where tan(θ) = 0 (i.e., multiples of π). Its range like tan(θ) covers the entire real number line.

Domain and Range of Secant Function {Sec (θ)}

The secant function, sec(θ) has a domain of all real numbers except the values where cos(θ) = 0. Its range is also (-∞, ∞) excluding values where cos(θ) = 0.

Domain and Range of Cosecant Function {Cosec (θ)}

The cosecant function, cosec(θ) shares its domain with the sine function, excluding the values where sin(θ) = 0 (i.e., multiples of π). Its range is (-∞, ∞) excluding values where sin(θ) = 0.

Domain and Range of Trig Functions Table

Here’s a summary table for the domain and range of trig functions:

Trigonometric Function	Domain	Range
sin(θ)	R	[-1, 1]
cos(θ)	R	[-1, 1]
tan(θ)	R excluding odd multiples of π/2	R
cot(θ)	R excluding multiples of π	R
sec(θ)	R excluding values where cos(x) = 0	R-[-1, 1]
cosec(θ)	R excluding multiples of π	R-[-1, 1]

Domain and Range of Inverse Trigonometric Functions

Inverse trigonometric functions are fundamental tools in calculus in the integration of trigonometric expressions. The behavior of inverse trigonometric functions helps in understanding the periodicity and symmetry inherent in trigonometric relationships.

These functions are essential in various fields including physics, computer graphics and navigation where understanding angles and rotations is essential.

The domain and range of inverse trigonometric functions such as sin^-1(x), cos^-1(x), tan^-1(x), cot^-1(x), sec^-1(x) and cosec^-1(x) are specific to ensure one-to-one correspondence between inputs and outputs.

Domain and Range of Inverse Trig Functions Table

The domain and range of the inverse of the trig functions added in the table are,

Inverse Trigonometric Function	Domain	Range
sin-1x	[-1, 1]	[-π/2, π/2]
cos-1x	[-1, 1]	[0, π ]
tan-1x	R	(-π/2, π/2)
cot-1x	R	(0, π)
sec-1x	R – (-1, 1)	[0, π] – {π/2}
cosec-1x	R – (-1, 1)	[-π/2, π/2] – {0}

Domain and Range of Trig Functions Using Graph

Domain and range of all the six trigonometric functions can also be found using the their graphs. The image with graphs of all six trigonometric function are added below,

By studying the image we can easily find the domain and range of the trigonometric functions.

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Examples on Domain and Range of Trigonometric Functions

Example 1: Find the domain and range of y = sin(x) for all real numbers.

Solution:

The domain and range of y will be as follows:

Domain: R

Range: [−1,1]

Example 2: Determine the domain and range of y = 2cos(3x) for all real values of x.

Solution:

The domain and range of y will be as follows:

Domain: R

Range: [−2,2]

Example 3: Determine the value of y = cot(2x) for x = 8π

Solution:

The value of y at x= 8π

= cot(2× 8π)

= 1

Example 4: Find the domain and range of y = −2tan(x)+1 for all real values of x.

Solution:

The domain and range of y will be as follows:

Domain: R

Range: (−∞, 1)

Example 5: Determine the value of y = cos( 2x) for x= 2π

Solution:

The value of y at x= 2π

= cos(4π)

= 2

Practice Problems on Domain and Range of Trigonometric Functions

Q1: Find the domain and range of y = cos(x) for all real numbers.

Q2: Determine the value of y = tan(x) for x= 2π.

Q3: Find the domain and range of y = −2sin(x)+1 for all real values of x.

Q4: Determine the value of y = sin(2x) for x = π.

Q5: Determine the value of y = sec(x) for x = 3π.

FAQs on Domain and Range of Trigonometric Functions

What is the Domain of a Trigonometric Function?

The domain of a trigonometric function is the set of all real numbers for which the function is defined and produces valid outputs.

What is the Range of a Trigonometric Function?

The range of a trigonometric function is the set of all possible output values that the function can produce.

How To Find Domain and Range of Trigonometric Functions?

To find the domain and range, consider the restrictions imposed by the function’s definition and the possible values of the trigonometric ratios.

What is the Domain and Range of Sine Function {Sin (x)}?

The domain of sin(x) is all real numbers and its range is between -1 and 1 inclusive: -1 ≤ sin(x) ≤ 1. The range of sine is [-1, 1].

Can a Trigonometric Function have an empty Domain or Range?

No, a trigonometric function does not have an empty domain as it is defined for all real numbers. However, its range is limited to specific values.

What is the Principal Range of Trigonometric Functions?

The principal range of trigonometric functions typically refers to the interval [-1, 1] for sine and cosine functions. For other trigonometric functions, the principal range may vary based on their definitions.

How to Find Domain and Range of Inverse Trigonometric Functions?

Domain and Range of Inverse Trigonometric Functions, is found by switching the domain and range of the original functions.