Trigonometric Form in Complex Numbers

It is essential to have the ability to convert complex numbers both from their rectangular form to their trigonometric form and from their trigonometric form to their rectangular form.

We use the following formula to convert a complex number, denoted by z, to its equivalent trigonometric form:

⇒ z = |z| (cos θ + i sin θ) —- (1)

In this case, the magnitude of z is denoted by |z|, whereas the phase of z is denoted by.

In this case, we have the equation 

z = 3 – 3i written out as z = a + bi, where a = 3 and b = -3.

Now, we know that |z| = √(a2 + b2) = √(32 + 32) = 3√2.

⇒ tan θ = b/a = -3/3 = -1 ⇒ θ = -π/4

Therefore, after making the necessary adjustments to the values of |z| and in equation (1),

 we have the result of 32 (cos(−π/4) + i sin(−π/4)).

The complex number 3 minus 3i is written in polar form as follows:

As a result, the complex number 3 – 3i can be written in the form of a trigonometric expression as 3√2 (cos(π/4) + i sin(π/4)).

How may complex numbers be converted into their trigonometric equivalents?

Let’s say that the complex number is written as z = (x+iy).

The polar form is written as (r,θ).

The formula for trigonometry is r equals cosθ+ sinθ.

Division Procedures for Complicated Numbers

Now that we understand what it means to divide complex numbers, 

let’s talk about the processes that are required to divide complex numbers. 

In order to divide the two complex numbers, carry out the following stages in order:

First, perform the calculation necessary to determine the conjugate of the complex number that is located in the fraction’s denominator.

Perform a multiplication using the complex fraction’s numerator and denominator, as well as the conjugate, respectively.

In the denominator, do the algebraic identity (a+b)(a-b)=a2 – b2 and then replace i2 with the value -1.

Simplify your calculations by using the distributive property in the numerator.

Distinguish between the real and the imaginary components of the complex number that was produced.

Division

It is difficult to divide a number by an imaginary number, which makes dividing complex numbers somewhat more difficult than adding, subtracting, and multiplying complex numbers.

 This is because it is difficult to divide a number by a real number. 

In order to divide complex numbers, we need to locate a term by which we may multiply the numerator and the denominator. 

This will allow us to remove the imaginary portion of the denominator and leave us with a denominator that contains a real value.

Important Remarks Regarding the Division of Complicated Numbers

Multiply the numerator and denominator of the fraction a+ib/c+id by cid, and then simplify the result. 

This will allow you to divide a complex number by another complex number.

The expression aib is the conjugate of the complex equation z = a+ib.

|z| is equal to √(a2 + b2), which is the modulus of the complex number z = a+ib.

Multiplication

The method of multiplying complex numbers involves applying the distributive property to multiply two or more complex numbers together. 

This can be done with any number of complex numbers. If we have two complex numbers, say 

z = a + ib and w = c + id,

 then the product of those two complex numbers can be expressed as 

zw = (a + ib) (c + id). 

This is an example of a mathematical expression. 

When figuring out the product of two complex numbers, we make use of the distributive property of multiplication.

The expression for writing a complex number in polar form is z = r (cos θ+ i sinθ ), where r is the modulus of the complex number and is the argument of the complex number. 

In polar form, the multiplication formula for the complex numbers

 z1 = r1 (cos θ1 + i sin θ1) and z2 = r2 (cos  θ2 + i sin θ2) is as follows:

z1z2 = [r1 (cos θ1 + i sin θ1)] [r2 (cos  θ2 + isin  θ2)]

= r1 r2 (cos θ1 cos  θ2 + i cos θ1 sin  θ2 + isin θ1 cos  θ2 + i2 sin θ1 sin  θ2)

= r1 r2 (cos θ1 cos  θ2 + i cos θ1 sin  θ2 + i sin θ1 cos  θ2 – sin θ1 sin  θ2) {Because i2 = -1}

= r1 r2 [cos θ1 cos  θ2 – sin θ1 sin  θ2 + i(cos θ1 sin  θ2 + sin θ1 cos  θ2)] 

= r1 r2 [cos (θ1 +  θ2) + i sin (θ1 +  θ2)] {Because cos a cos b – sin a sin b = cos (a + b) and sin a cos b + sin b cos a = sin (a + b)}

As a result, the formula for multiplying complex numbers when they are written in polar form is [r1 (cosθ1 + i sinθ1)]. [r2 (cos  θ2 + i sin  θ2)] = r1 r2 [cos (θ1 +  θ2) + i sin (θ1 +  θ2)]

Moduli

The term moduli is a plural version of modulus, and the magnitude of a number can be determined using a modulus function regardless of the sign of the number. 

A different name for this concept is the absolute value function. In mathematics, the modulus of a real number x is determined by using a function called the modulus function, which is symbolised by the notation |x|.

 It provides the value of x that is not negative. The distance that a number is from its starting point, which is represented by zero, can be thought of as the modulus of the number or its absolute value.

Conclusion

A complex number is a number that takes the form a+bi, where a and b are both real numbers and i is the imaginary unit, or 1. 

Complex numbers can only be formed using this format. 

Complex numbers and imaginary numbers, despite their nomenclature, have extremely practical and significant applications in mathematics as well as in the real world. 

These applications can be found in both places. 

Complex numbers are helpful in pure mathematics because they offer a number system that is more consistent and versatile, which makes it easier to solve issues involving algebra and calculus.