Trigonometric Functions – Domain and Range

Sin, cos, tan, cot, tan, 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 number. Depending on the region where the trigonometric function is not defined, some values are excluded from the domain and range of trigonometric functions. 

What exactly are the domain and range of trigonometric functions?

The input and output values of trigonometric functions are the domain and range of trigonometric functions, respectively. The domain of trigonometric functions denotes the values of angles where the trigonometric functions are defined, whereas the range of trigonometric functions denotes the resultant value of the trigonometric function corresponding to a specific angle in the domain. There are six basic trigonometric functions: sin, cos, tan, cot, tan, cosec, and sec.

Trigonometric Function Domain and Range: Sine

We know that the sine function is the ratio of a right-angled triangle perpendicular and hypotenuse. The trigonometric function sine’s domain and range are given by:

• Domain = All natural numbers
• [-1, 1] is the range.

Trigonometric Function Domain and Range: Cosine

The cosine function is defined as the ratio of the adjacent side and hypotenuse of a right-angled triangle. The trigonometric function cosine’s domain and range are given by:

• Domain = All natural numbers
• [-1, 1] is the range.

Domain and Range of Trigonometric Function: Tangent

The tangent function is defined as the ratio of the opposite and adjacent sides of a right-angled triangle. Because it can also be written as the ratio of the sine and cosine functions, the domain of tan x does not contain values where cos x is equal to zero. Because cos x is 0 at odd integral multiples of π/2, the domain and range of the trigonometric function tangent are given by:

• Range = all real numbers
• Domain = R – (2n + 1) π/2

Trigonometric Function Domain and Range: Cotangent

The cotangent function is defined as the ratio of the adjacent and opposite sides of a right-angled triangle. It is also known as the cosine-sine function ratio, and cot x is the reciprocal of tan x. As a result, the domain of cot x does not contain any values where sin x equals zero. Because sin x is 0 at integral multiples of, the domain and range of the trigonometric function cotangent are given by:

• Domain = R – nπ
• Range = (−∞, ∞)

Domain and range of a trigonometric function: Secant

In a right-angled triangle, the secant function is defined as the ratio of the hypotenuse to the adjacent side. It is also known as the cosine function reciprocal. As a result, the domain of sec x does not contain any values where cos x equals zero. Because cos x is 0 at odd integral multiples of, the domain and range of the trigonometric function secant are given by:

• Domain = R – (2n + 1)π/2
• Range = (-∞, -1] U [+1, +∞)

Trigonometric Function Domain and Range: Cosecant

The cosecant function is defined as the ratio of the hypotenuse to the opposite side of a right-angled triangle. It is also known as the sine function reciprocal. As a result, the domain of the trigonometric function cosec x does not contain any values where sin x equals zero. Because sin x is 0 at integral multiples of, the domain and range of the trigonometric function cosecant are given by:

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

Table of Trigonometric Functions Domain and Range

We’ve now looked at the domain and range of trigonometric functions. The table below summarizes it, which will aid in better understanding and application to various problems:

Inverse Trigonometric Functions’ Domain and Range

If and only if a function is bijective, it is invertible. The inverse trigonometric functions are the inverse of the trigonometric functions, and we restrict the domains of the trigonometric functions to the principal value branch to make them invertible. The domain and range of the inverse trigonometric functions are represented in the table below:

Points to Remember

• In cases where the function is not defined, look for the value of input. The domain can be excluded from the value where the function is not defined.
• The output values for each of the input values define the range of a trigonometric function (domain).

Conclusion 

Angles in degrees or radians are the domain of trigonometric functions, with a real number as the range. According to the region where the trigonometric function is not defined, some values are excluded from the domain and range of trigonometric functions. A function is nothing more than a rule that is applied to the values that have been entered. The domain of the function refers to the set of values that can be utilised as inputs to the function. As x fluctuates within the domain, the range is the set of possible values for the dependent variable.