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Tangent Function: Domain, Range, Properties and Applications


21 December  2020                

Read time: 5 minutes


Mathematics is a subject that is dynamic for gaining a better perspective on events that happen in the natural world.

A keen aptitude for math improves critical thinking and promotes problem-solving abilities.

One distinct/special area of mathematical and geometrical reasoning is trigonometry which studies the properties of triangles.

Now it's true that triangles are one of the simplest geometrical figures, yet they have varied applications. 

triangle of trigonometry with sides a,b and c

Trigonometry comes from Greek trigono "triangle" + metron "measure".

Trigonometry defines relationships between side lengths and angles of triangles.

Trigonometry also helps us find angles and distances. It is all about triangles. 

Also read,

What is the Tangent Function?

We will define the tangent function or tangent ratio of an angle as the ratio of the length of the opposite side to the length of the adjacent side.

trigonometric ratios, tangent and unit circle

Tan full form: The full form of the tan is tangent.

The tangent function is denoted by tan(x) . It is one of the six common trigonometric functions. Tangent is always almost related to sine and cosine.
The tangent ratio is as follows:
               Tan = Opposite/Adjacent = CB/BA
 Also, in terms of sine and cos, tangent ratio can be represented as:
               tan x = sin x/cos x
        or, tan theta = sin theta/cos theta   where theta is an angle 
We know that the sine of an angle is equal to the length of the opposite side divided by the length of the hypotenuse side whereas the cosine of the angle is the ratio of the length of the adjacent side to the ratio of the hypotenuse side.
That is, sin x = Opposite Side/ Hypotenuse Side
            cos x = Adjacent Side/ Hypotenuse Side
Therefore,  tan x = Opposite Side/ Adjacent Side

Example 1



What is the tangent of 30°?

30deg triangle: calculate tan30

Solution: In a 30° triangle hypotenuse of length 2, an opposite side of length 1 and an adjacent side of √3:

Tangent =  tan(30°) = 1 / 1.732 = 0.577



Example 2



What is the tangent of 35°?

35 deg triangle, calculate tan35

Solution: In a 35° triangle, an opposite side of length 2.8 and an adjacent side of 4:

Tangent = tan (35°) = 2.8/4.0 = 0.70      


Example 3



What is the tangent of 45°?

45 deg triangle: tan45 value

Solution: In a 45° triangle there is an opposite side of length 1 and an adjacent side of 1:

Tangent = tan(45°) = 1 / 1 = 1

Basic Properties of Tan Function 

Even and Odd functions

Cosine function and Sec functions are even functions; the remaining other functions are odd functions. Hence, the tan function is odd.
f(-x) = f(x)......................Even Function
f(-x) = -f(x).....................Odd Function

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

At some angles the tangent function is undefined, and the problem is fundamental to drawing the graph of tangent function.

Finding all values of x on the interval [0,2π] such that tan⁡(x) is undefined,
We start by using the definition of the tangent to rewrite it as
     tan(x) = sin(x) / cos(x)

The fraction is undefined where the denominator is 0, so we wish to solve the equation

In the given domain, the solutions are x=π/2 and x=3π/2, according to the arccosine function.

Graph of Tan Function

The tangent of an angle is designed against that angle measure to produce the tan graph.
Given, an example:
tan (52°) = 8.2/6.5 = 1.8. Placing trigonometric values like this against the angle produces a tan graph.

tan graph: Graph of tangent function

The tan graph is the graphical representation of the function tan x.

It is a periodic graph whose trigonometric values can be computed using the trigonometric formula: sin/cos=tan

Using this trigonometric formula, we realize all the points where cos x is 0, the tan x value is undefined.

Hence we get a smooth curve with the value of tan tending to infinity every multiple of 90 degrees or 3pi/2. 

Range of tangent, Period of tangent and other Properties

Tangent Function : tan(x)

tangent function graph: domain and range

Domain of Tangent

 All real numbers (R) except pi/2 + k pi, k is an integer. R-{pi/2 + kpi} k is in Z

Range of Tangent

All real numbers

Period of Tangent



x = k pi, place k is an integer.


y = 0


since tan(-x) = - tan(x) then tan (x) is an odd function and the graph of tanx is symmetric with respect to the origin.

Intervals of increase/decrease

Over one period and from -pi/2 to pi/2, tan(x) is increasing.

Vertical asymptotes

x = pi/2 + k pi, where k is an integer are the vertical asymptotes for a tangent graph. 

Range of Tangent

The range of a function is the set of result values it can produce. The tangent function has a range that goes automatically from positive infinity to negative infinity.

Applications of Trigonometric Ratios in Real Life

Can trigonometry be used in everyday life?

Trigonometry has not its direct applications in solving practical issues, but it is used in various things that we enjoy so much.

Sound travels in the form of waves, and the same pattern though not as regular as a sine or cosine function, is still useful in developing computer music.

A computer cannot listen and comprehend music as we do, so computers represent it mathematically by its constituent sound waves.

Mentioning some technological fields where there’s extensive use of trigonometric concepts.

  • Astronomy,
  • Navigation,
  • Optics,
  • Acoustics (The science of studying mechanical waves in solids, liquids and gases that also topics like sound, infrasound, ultrasound, and vibration),
  • Electronics,
  • Statistics,
  • Number theory,
  • Electrical engineering,
  • Mechanical engineering,
  • Computer graphics,
  • Game development,
  • Civil engineering,
  • Medical imaging,
  • Pharmacy,
  • Cartography (creating maps),
  • Seismology (It’s the science of studying earthquakes),
  • Crystallography (The study of atom arrangements in a crystalline solid).


Trigonometry has a variety of applications ranging from specialized fields like oceanography where it is used for calculating the height of tides in oceans and Calculus where it is used in combination with Algebra to the backyard of our home where it may be used to roof a house, to make the roof inclined in the case of single individual bungalows and to calculate the height of the roof in the buildings, etc.

The tangent function is also known as tan x, tan theta, or tan function is basically one of the 6 trigonometric functions. Tanx is the ratio of the length of the opposite to the length of the adjacent. 

Also read,

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Frequently Asked Questions (FAQs)

What is tan theta or tan function?

tan theta or tangent of an angle is the ratio of the length of the opposite to the length of the adjacent.

Value of tan 0?

The value of tan 0 degrees is 0

Value of tan 30?

The value of tan 30 degrees is 1/√3

Value of tan 45?

 The value of tan 45 degrees is 1

Value of tan 60?

The value of tan 60 degrees is √3

Value of tan 90?

The value of tan 90 degrees is Undefined

What is the domain of tangent?

All real numbers (R) except pi/2 + k pi, k is an integer. R-{pi/2 + kpi} k is in Z

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