## Inverse Cot(x)

Inverse cot x is one of the basic trigonometric inverse functions. It is also known by different names such as arc cot x, inverse cot, and inverse cotangent. Cot inverse x is the inverse function of the trigonometric function cot x and is written cot-1x. Please note that the cover is not a replacement for the pollution service. Cot inverse x gives the measure of the side of a right triangle equal to the given measure of the base and side.

Let’s review the inverse trigonometric function inverse cot x, its formula, its domain and limit, derivation and related graphs. We will also solve some examples based on the fabric for better understanding.

## What is the Inverse Cot of x?

Cot Inverse x is an inverse trigonometric function that gives the angle measurement in radians or degrees corresponding to the value of x.

In mathematics, it is written cot^{-1}x or arccot x, called ‘inverse cot x’ and ‘arc cot x’ respectively. If the function f is invertible and its inverse is f^{-1}, then we have f(x) = y ⇒ x = f^{-1}(y). So we can have the inverse cot x, if x = cot y, then we have y = cot^{-1}x. The function of the inverse cot x is as follows:

- If cot π/2 = 0, then cot
^{-1}0 = π/2 - If cot π/6 = √3, then cot
^{-1}√3 = π/6 - If cot π/3 = 1/√3, then cot
^{-1}1/√3 = π/3. - If cot π/4 = 1, then cot
^{-1}1 = π/4

If you are still confused, try the Inverse cotangent calculator available online to do this type of calculation.

## Reverse Cot x Formula

We know that in a right triangle we have cot θ as the ratio of one side to another, i.e. cot θ = side angle/front angle. So, using the definition of inverse cot x, we have θ = inverse cot which is side/side, so the formula for the inverse cot is:

**θ = cot ^{-1}(adjacent side/opposite side)**

## Binomial theorem

The binomial theorem helps to find the expanded value of an algebraic expression of the form (x + y)^{n}. Finding the value of (x + y)^{2}, (x + y)^{3}, (a + b + c)^{2} is easy and can be achieved by using algebra to multiply the number of times by the value of the exponent. But finding the expanded form of (x + y)^{17} or other similar expressions with higher explanatory values involves a lot of calculations. This can be simplified using the binomial theorem.

The exponential value of the binomial linear expansion can be a negative number or a fraction. Here, we limit our interpretation to negative values only. Let’s learn more about the structure, process, and properties of numbers in this binomial expansion story.

## What is the Binomial Theorem?

The first mention of the binomial theorem dates back to the fourth century BC by a famous Greek mathematician named Euclid. The binomial theorem states the principle of the expansion of the algebraic expression (x + y)n and is expressed as the sum of the terms related to each variable x and y. Each term of the binomial expansion is associated with a numerical value called a coefficient.**Statement:** According to the binomial theorem, it is possible to expand any negative power of the binomial (x + y) into sums of the form,**(x+y) ^{n} = ^{n}C_{0 }x^{n}y^{0} + ^{n}C_{1 }x^{n-1}y^{1} + ^{n}C_{2 }x^{n-2}y^{2}+…+ ^{n}C_{n-1}x^{1}y^{n-1} + ^{n}C_{n }x^{0}y^{n}**

where, n ≥ 0 is an integer and every nCk is a positive number called a binomial number.

Note: When the exponent is equal to zero, the corresponding power term is 1. The derivative is often left out of the expression, so the right-hand side is often written as nC0 xn + …. This formula is also called the binomial formula or the binomial identity. Using summation, the binomial theorem can be given by,**(x+y)n = ∑ ^{n}k=0^{n}C_{k} x^{n-k}y^{k} = ∑^{n}k = 0^{n}C_{k} x^{n-k}y^{k}** Expand (x+3)5 using the binomial formula. Here y = 3 and n = 5. Substituting and expanding, we get:

Example:

(x+3)

^{5}=

^{5}C

_{0}x

^{5}3

^{0}+

^{5}C

_{1}x

^{5-1}3

^{1}+

^{5}C

_{2}x

^{5-2}3

^{2}+

^{5}C

_{3}x

^{5-3}3

^{3}+

^{5}C

_{4}x

^{5-4}3

^{4}+

^{5}C

_{5}x

^{5-5}3

^{5}

= x

^{5}+ 5x

^{4}. 3 + 10x

^{3}. 9 + 10x

^{2}. 27 + 5x .81 + 3

^{5}

= x

^{5}+15x

^{4}+90x

^{3}+270x

^{2}+405x+243

Moreover, you may also try the Binomial Coefficient Calculator for doing such complex calculations.

## Binomial expansion

The binomial theorem is also known as the binomial expansion which provides a method for the expansion of the exponential power of the binomial expression. The binomial expansion of (x + y)n using the binomial theorem is,**(x+y) ^{n} = ^{n}C_{0 }x^{n}y^{0} + ^{n}C_{1 }x^{n-1}y^{1} + ^{n}C_{2 }x^{n-2}y^{2}+…+ ^{n}C_{n-1}x^{1}y^{n-1} + ^{n}C_{n }x^{0}y^{n}**

## The binomial sequence of events

The binomial theorem is used in all binomial expansions as a function. The formula for the binomial theorem is (a+b)n= ∑nr=0nCr an-rbr, where n is a real integer and a, b are real numbers, and 0 < r ≤ n. This method helps to expand binomial expressions like (x + a) 10, (2x + 5) 3, (x – (1/x)) 4, etc. The binomial function method helps in the expansion of a binomial raised to a certain power.

Let us understand the binomial theorem and its application in the following sections. The binomial theorem states: If x and y are real numbers, then for all n ∈ N,

**(x+y) ^{n} = ^{n}C_{0 }x^{n}y^{0} + ^{n}C_{1 }x^{n-1}y^{1} + ^{n}C_{2 }x^{n-2}y^{2}+…+ ^{n}C_{n-1}x^{1}y^{n-1} + ^{n}C_{n }x^{0}y^{n}**

**⇒ (x + y)**

where, nCr = n! / [r! (n-r)!]

^{n}= ∑^{n}k = 0^{n}C_{k}x^{n-k}y^{k}So these are some identical differences between these topics. Moreover for more educational blogs, you may visit e-tv UK.