Tangent Function
The tangent function is one of the basic trigonometric functions and is quite a commonly used function in trigonometry. The tangent function can be expressed as the ratio of the sine function and cosine function. In a rightangled triangle, the formula for the tangent function is expressed as the ratio of the perpendicular and base of the triangle. It can also be expressed as the reciprocal of the cotangent function. Mathematically, tan function is written as f(x) = tan x
Further in this article, we will explore the tangent function graph, its domain and range, the trigonometric identities of tan x, and the formula of the tangent function. We will also solve some examples related to the tan function for a better understanding of the concept.
1.  What is Tangent Function? 
2.  Tangent Function Graph 
3.  Domain and Range of Tangent Function 
4.  Properties of Tangent Function 
5.  FAQs on Tangent Function 
What is Tangent Function?
The tangent function is one of the main six trigonometric functions and is generally written as tan x. It is the ratio of the opposite side and the adjacent side of the angle in consideration in a rightangled triangle. We have various trigonometric identities and formulas related to the tangent function that can be derived using different formulas. The formula for the period of the tangent function f(x) = a tan (bx), is given by, Period = π/b. Tangent function tan x is a periodic function and has a period of π/1 = π (Because b =1 in tan x).
Tangent Function Formula
Now, we have two main formulas for the tangent function. As we know that, in a rightangled triangle, tan x is expressed as the ratio of the opposite side and the adjacent side of the angle in consideration. The tangent function can also be expressed as the ratio of the sine function and cosine function which can be derived using a unit circle. Hence, the formulas for tan x are:
 tan x = sin x/cos x
 tan x = Opposite Side/Adjacent Side = Perpendicular/Base
Tangent Function Graph
The graph of the tangent function is a discontinuous graph as the value of tan x is not defined at odd multiples of π/2, that is, tan x is not defined for x = kπ/2, where k is an odd integer. Also, since the tangent function has a period of π, therefore its values repeat after every π radians and hence, the pattern of the curve is repeated after every π radians. As we can see in the graph of the tangent function given below, the function has vertical asymptotes at x = π/2, π/2, 3π/2, 3π/2, 5π/2, ....
Domain and Range of Tangent Function
The tangent function is not defined at odd multiples of π/2 as the length of the base in a right triangle is 0 and cos x = 0 when x = kπ/2, where k is an odd integer. Hence, the domain of tan x is all real numbers except the odd multiples of π/2. Now, the range of the tangent function includes all real numbers as the value of tan x varies from negative infinity to positive infinity. Therefore, we can conclude:
 Domain = R  {(2k+1)π/2}, where k is an integer.
 Range = R, where R is the set of real numbers.
Properties of Tangent Function
Next, let us go through some of the important properties of the tangent function. The basic properties of tan x along with its value at specific angles and the trigonometric identities involving tan x are:
 The tangent function is an odd function because tan (x) = tan x.
 Tan x is not defined at values of x where cos x = 0.
 The graph of tan x has an infinite number of vertical asymptotes.
 The values of the tangent function at specific angles are:
 tan 0 = 0
 tan π/6 = 1/√3
 tan π/4 = 1
 tan π/3 = √3
 tan π/2 = Not defined
 The trigonometric identities involving the tangent function are:
 1 + tan^{2}x = sec^{2}x
 tan 2x = 2 tan x/(1  tan^{2}x)
 tan (a  b) = (tan a  tan b)/(1 + tan a tan b)
 tan (a + b) = (tan a + tan b)/(1  tan a tan b)
 The graph of tan x is symmetric with respect to the origin.
 The xintercepts of tan x are where sin x takes the value zero, that is, when x = nπ, where n is an integer.
Important Notes on Tangent Function:
 The tangent function is expressed as tan x = sin x/cos x and tan x = Perpendicular/Base
 The slope of a straight line is the tangent of the angle made by the line with the positive xaxis.
☛ Related Topics:
Tangent Function Examples

Example 1: Find the value of the tangent function in a rightangled triangle when the adjacent side = 3 units, opposite side = 4 units, and hypotenuse = 5 units.
Solution: Using the formula of the tangent function, we have
tan x = opposite side/adjacent side
= 4/3
Answer: tan x = 4/3

Example 2: Find the exact length of the shadow cast by a 15 ft tree when the angle of elevation of the sun is 60º.
Solution: The height of the tree = 15 ft = Perpendicular
Assume the length of the shadow = x = Base
Angle of elevation = 60°
Therefore, using the formula of the tangent function, we have
tan 60° = 15/x
⇒ √3 = 15/x
⇒ x = 15/√3
= 5√3 ft.
Answer: The length of the shadow is 5√3 ft.
FAQs on Tangent Function
What is Tangent Function in Trigonometry?
The tangent function is one of the basic trigonometric functions and is quite a commonly used function in trigonometry. The tangent function can be expressed as the ratio of the sine function and cosine function. It is the ratio of the opposite side and the adjacent side of the angle in consideration in a rightangled triangle.
What is the Range of the Tangent Function?
The range of the tangent function includes all real numbers as the value of tan x varies from negative infinity to positive infinity. Hence, the range is equal to the set of all real numbers R.
What is the Period of the Tangent Function?
The formula for the period of the tangent function f(x) = a tan (bx) is given by, Period = π/b. Tangent function tan x is a periodic function and has a period of π/1 = π (Because b =1 in tan x)
How to Find Domain of Tangent Function?
The tangent function is not defined at odd multiples of π/2 as the length of the base in a right triangle is 0 and cos x = 0 when x = kπ/2, where k is an odd integer. Hence, the domain of tan x is all real numbers except the odd multiples of π/2. Therefore, the domain of the tangent function tan x is R  {(2k+1)π/2}, where k is an integer.
How to Find xintercept of Tangent Function?
The xintercepts of tan x are where sin x takes the value zero, that is, when x = nπ, where n is an integer. Hence, the xintercepts of tan x are at x = nπ.
Why is Tan Called Tangent?
The abbreviated form of the tangent function is tan x. Therefore, tan is called tangent function in expanded form.
What is Tangent Function Formula?
We have two main formulas for the tangent function. The formulas for tan x are:
 tan x = sin x/cos x
 tan x = Opposite Side/Adjacent Side = Perpendicular/Base
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