Tangent Function: Formula, Graph, Identities, Domain and Range

In a triangle with three sides, there are six combinations of ratios of sides of the right-angled triangle. Tangent is one of them. There are many trigonometric formulas related to the tangent function which are obtained through different formulas. Tan function is expressed as f(x) = tan x. The graph of the tan function has an infinite number of vertical asymptotes.

Also read: Domain and Range of Trigonometric Functions

Key Takeaways: Tangent, Trigonometry, Sine, Cosine, Domain, Tan, Right-angled triangle, Right angle, Angle, Adjacent side

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Definition of Tangent

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Tangent is the ratio of the opposite side and the adjacent side of the angle θ (angle in consideration) in a right-angled triangle. The opposite represents the side opposite the reference angle θ, and the adjacent represents the side between the angle θ and the right angle. 

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Tangent Function Formula

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There are two key formulas for the tangent function. In a right-angled triangle, tan x is represented 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.

tan x = sin x/cos x

tan x = Opposite Side/Adjacent Side OR tan x= Perpendicular/Base

 tan function using unit circle

Derivation of Tangent Function

Also read: Introduction to Trigonometry

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Tangent Function Graph

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• The tangent graph is a visual representation of the tangent function for a given range of angles. 
• The horizontal axis(x-axis) of a trigonometric graph represents the angle, written as theta (θ), and the vertical axis (y-axis) is the tangent function of that angle. 
• The tangent function has a discontinuous graph as the value of tan x is not defined at odd multiples of π/2. So, tan x is not defined for x = kπ/2, where k is an odd integer. 
• The tangent function has a period of π; thus, its values repeat after every π radians and so, the pattern of the curve is repeated after every π radians. As we can see in the graph of the tangent function at x = -π/2, π/2, -3π/2, 3π/2

Tangent graph

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Tangent Function Values

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Also read: Trigonometry Values: Identities, Ratios and Formula

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Tangent Function Identities

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• Identities

Below are the identities related to trigonometric functions.

Even and Odd functions

The cos and sec functions are even functions; the rest other functions are odd functions.

• sin(-x) = -sin x
• cos(-x) = cos x
• tan(-x) = – tan x
• cot(-x) = -cot x
• csc(-x) = -csc x
• sec(-x) = sec x

Periodic Functions

The trigonometric functions are the periodic functions. The smallest periodic cycle is 2π but for tangent and the cotangent it is π.  

• sin(x+2nπ) = sin x
• cos(x+2nπ) = cos x
• tan(x+nπ) = tan x
• cot(x+nπ) = cot x
• csc(x+2nπ) = csc x
• sec(x+2nπ) = sec x

Where n is any integer.

Pythagorean Identities

When the Pythagoras theorem is expressed in the form of trigonometry functions, it is said to be Pythagorean identity. There are majorly three identities:

• sin² x + cos² x = 1    [Very Important]
• 1+tan² x = sec² x
• cosec² x = 1 + cot² x

These three identities are of great importance in Mathematics, as most of the trigonometry questions are prepared in exams based on them. Therefore, students should memorise these identities to solve such problems easily.

Sum and Difference Identities

• sin(x+y) = sin(x).cos(y)+cos(x).sin(y)
• sin(x–y) = sin(x).cos(y)–cos(x).sin(y)
• cos(x+y) = cosx.cosy–sinx.siny
• cos(x–y) = cosx.cosy+sinx.siny
• tan(x+y) = [tan(x)+tan(y)]/[1-tan(x)tan(y)]
• tan(x-y) = [tan(x)-tan(y)]/[1+tan(x)tan(y)]

Also read: Introduction to Trigonometry Formula

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Domain and range of Tangent Function

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 If the length of the base in a right triangle is 0, cos x = 0 (when x = kπ/2, where k is an odd integer). Therefore, the domain of tan x is all real numbers except the odd multiples of π/2. 

Domain = R - {(2k+1) π/2}, where k is an integer

The range of the tangent function includes all real numbers since the value of tan x varies from negative infinity to positive infinity. 

Range = R [R: set of real numbers]

Also read: Sin Cos Formulas: Basic Trigonometric Ratios and Identities

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Tangent Function Properties

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The basic properties of tan function along with its value at specific angles and the trigonometric identities:

• The tangent function is an odd function because tan (-x) = -tan x.
• Tan x function 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 values of angles are: