# What are Transcendental Functions?

*I am a PhD student of mathematics. I have complete MS in math from the University of Pakistan and have been writing online since 2020.*

## Transcendental Functions

**Outline**

- Introduction
- Examples
- Graphical Behavior of Sin (r)
- Graphical Behavior of cos (r)
- Graphical Behavior of tan (r)
- Graphical Behavior of e
^{-}x - The graphical representation of e
^{x} - The graphical representation of e power -x square (e^ (-x^2))
- The graphical representation of e power x square (e^(x^2))
- FAQs
- References
- Related Links

**Introduction:**

The definition of transcendence is to go past a limit or a boundary. A function in mathematics that cannot be expressed by combining addition, subtraction, multiplication, division, raising to a power, and taking the root. The functions log, sin, cos, and e^{x}, as well as any functions containing them, are examples. The exponential function, trigonometric functions, and their inverses are all examples of transcendental functions. First, according to Euler, some operations are transcendental. He referred to the following operations as algebraic even though they took place in a section discussing functions: addition, subtraction, multiplication, division, raising to a power, and the extraction of root. The algebraic functions are those which can be written in the form of algebraic expression. The transcendental functions are those which are not algebraic. Some functions are algebraic, but some are transcendental.

**Examples of Transcendental functions:**

There are three types of transcendental functions like,

- Trigonometric functions
- Exponential functions
- Inverse of trigonometric and exponential functions

**Trigonometric Functions:**

There are 3 basic trigonometric functions, sin(r), and cos(r), tan(r). The other three trigonometric functions like, sec (r), cosec(r) and cot (r) are made by three basic trigonometric functions. In this section we will discuss the graphs of sin(r), and cos(r), tan(r) in 3-dimension.

**Graphical Behavior of Sin (r):**

The graphical representation of transcendental function sin (r) in 3-dimensional is shown in figure no.1 and figure no.2.

- The sin curve is sinusoidal.
- The behavior of sin curve is decreasing and increasing.

**Graphical Behavior of cos (r):**

The graphical representation of transcendental function cos (r) in 3-dimensional is shown in figure no.3 and figure no.4.

- The cos curve is initially increase then decrease and with the passage of time it again increases.

**Graphical Behavior of tan (r):**

The graphical representation of transcendental function tan (r) in 3-dimensional is shown in figure no.5 and figure no.6.

The tan function is a ratio of cos and sin function i.e.

Tan (r) = sin(r) / cos(r)

So, the curve of tan is the combination of curve of sin as well as cos.

- The tan curve is initially constant then increase at the same point it is increase.
- Then it again become constant then increase at the same point it decrease and at the end it again become constant.

**Exponential functions:**

Exponential functions are transcendental functions like, **e ^{-x}**,

**e**etc.

^{x }In this section we will discuss the graphical behavior of exponential functions in 3-dimensuinal.

**Graphical Behavior of e^{-}**

**x:**

The graphical representation of** e ^{-x}**is shown in figure no.7.

**The graphical representation of e ^{x}**

The graphical representation of e^{x }is shown in figure no.8.

**The graphical representation of e power -x square (e^(-x^2))**

The graphical representation of e power -x square (e^(-x^2)) is shown in figure no.9.

**The graphical representation of e power x square (e^(x^2)):**

The graphical representation of e power x square (e^(x^2)) is shown in figure no.10.

**FAQs:**

**1. What are transcendental functions?**

A function in mathematics that cannot be expressed by combining addition, subtraction, multiplication, division, raising to a power, and taking the root.

** 2. How many types of transcendental functions?**

there are three types of transcendental functions discussed below,

- Trigonometric Functions
- Exponential Functions
- Inverse of trigonometric functions and exponential functions.

**3. What is the behavior of curve of transcendental function cos (r)?**

The cos curve is initially increase then decrease and with the passage of time it again increases.

**4. What is the behavior of curve of transcendental function tan (r)?**

The tan function is a ratio of cos and sin function i.e.

Tan (r) = sin(r) / cos(r)

So, the curve of tan is the combination of curve of sin as well as cos.

- The tan curve is initially constant then increase at the same point it is increase.
- Then it again become constant then increase at the same point it decreases and at the end it again become constant.

**5.What is the behavior of curve of transcendental function sin (r)?**

- The sin curve is sinusoidal.
- The behavior of sin curve is decreasing and increasing.

** **

**References:**

Larson, R. and Edwards, B.H., 2010. *Calculus: Early transcendental functions*. Cengage Learning.

**Related Links:**

- Applications of Trigonometric Functions

Trigonometry is used in every field of life. Here we will discuss the applications of trigonometry in the field of medicine, engineering, chemistry, broadcasting, development of gaming as well as in seismology.

- Exponential Functions

Exponential functions are used in our daily life. it is an important topic of mathematics. Here we will learn the definition, basic properties, examples and solved problems of exponential functions.

- Trigonometric Functions

The most significant mathematical functions are trigonometric functions. We will learn about its definition, key formulas, and relevant formulas and examples for trigonometric functions and their graphs.

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

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