Domain and Range of Trigonometric Functions

Every mathematical function is designed to work with different kinds of values. A function y=f(x) takes certain input values to give corresponding output values. Not all functions take the same values to give output. 

The unique set of values that a function takes as an input is its domain. Similarly, the function’s value or output for a given domain is its range.

In simple terms, a function generates a range of values for numbers in the given domain. Both domain and range are unique for a function. In this article, we will study the domain and range of trigonometric functions.

What are Trigonometric Functions?

Trigonometric functions are functions of angles in a triangle. These functions give a relationship between the angles and the sides of a triangle. They take angles as their input and generate values according to those measurements. 

We know about three fundamental trigonometric functions. Consider a right-angled triangle with an angle A. This setup is shown in a circle where if any point P is on the circle and is connected to the center C of the circle, it forms a right-angled triangle.

The various trigonometric ratios and their inverse are 

•     sine A or sin A=1/cosec A
•     cos A or cosine A=1/sec A
•     tan A or tangent A=1/cot A
•     cosec A= cosecant A=1/sin A
•     sec A= secant A= 1/ cos A
•     cot A= cotangent A =1/tan A

Corresponding Trigonometric functions are:

•     f(x)= sin(x)
•     f(x)= cos(x)
•     f(x)= tan(x)
•     f(x)= sec(x)
•     f(x)= cosec(x)
•     f(x)= cot(x)

Domain & Range of Various Trigonometric Functions

We will now consider all of the above six trigonometric functions and find out their domain, i.e., the values of x for which the function holds good. 

y=f(x) =sin(x)

The function sin(x) is defined as the opposite side of angle x divided by the hypotenuse. For any point in a unit circle, sin(x) equals opposite / 1 . With the same measure, all the points on the circle can be defined. This means that angle x can take any value. 

So, the function y=sin(x) has the domain of all real numbers.

For the same function sin(x), its value depends on the point on the circle. It can go to a maximum of 1 when x becomes 90 degrees and a minimum of -1 when x becomes 270 degrees. This gives a range of -1 to 1.

So, for y = sin(x):

Domain D = R[+ ∞, – ∞]

Range = [-1, +1]

y=f(X) = cos(x)

Like sin(x), for cos(x), the angle x can also take any value as per the position of the point P on the circle. As we already know, cos(x) = adjacent /1.

So, the function cos(x) also has all real numbers in its domain.

To find the function cos(x) range, you need to know where the point on the circle is. It can start from a minimum of -1 when x becomes equal to 180 degrees and go to a maximum of 1 when x becomes equal to 0 degrees.

So, for y = cos(x):

Domain D= R [- ∞, + ∞]

Range = [- 1, + 1]

y=f(x) = tan(x)

You can consider tan x = sin x / cos x. If you now analyze the value of this function, tan x will be defined for all values except when the denominator cos x becomes 0, which is not defined.

Now cos x=0 for every odd multiple of 90 or π/2. 

So, Domain D of tan(x) is all real numbers except when cos (x) is 0, i.e., values π/2+πn for all integers n. 

D= R – (2n + 1) π/2

The Range R of tan(x) is all real numbers or [- ∞, + ∞].

y=f(X) = cosec(X)

The function y=f(x) = cosec(x) = 1/sin(x) is all real numbers except any value that makes the denominator 0 when it becomes undefined. This means that all those values when sin(x) =0, are excluded from its domain. 

So, Domain cosec(x) = R-nπ. This covers all nπ values for all integers n.

Range of cosec(x) =R-(-1,1)

y=f(X)= sec(X)

The function y=f(x) = sec(x) = 1/cos(x) is all real numbers except any value that makes the denominator 0 when it becomes undefined. This means that all those values when cos(x) =0, are excluded from its domain. 

So, Domain sec(x) = R-(π/2+nπ). This covers all nπ values for all integers n.

Range of sec(x) =R-(-1,1)

y=f(X) =cot(X)

The function y=f(x) = cot(x) is 1/tan(x) = cos(x)/ sin(x). This means its domain is all real numbers except any value when the denominator becomes 0 or undefined. Its domain does not include all those values when sin(x) =0 which is for x=nπ.

So, Domain cot(x) = R- nπ, for all integers n.

The range of function cot(x) is all real numbers.

Conclusion

You can easily find the domain and range of the basic trigonometric functions once you know how to calculate their values. You can even use a trigonometry calculator for this purpose. The trigonometry calculator is an online tool where you just need to enter the required given values to find the values of trigonometric formulas for sin(x), cos(x), and tan(x) and their reciprocal functions.