# Complex Numbers: Expressing (1- i tanx) / (1+ i tanx) in Polar Form

• abcd8989
In summary, to express (1- i tanx) / (1+ i tanx) in polar form, first multiply the fraction by cosx to get (cosx - i sin x) / (cosx + i sin x). The modulus is 1 and the argument is -2x. Another way to handle it is to multiply by (cosx - i sin x) / (cosx - i sin x), resulting in (cos2x-sin2x-2cosxsinx i). The modulus is still 1, but the argument is harder to find and can be written as tanB=-tan2x. However, this method loses track of the signs of a and b separately

#### abcd8989

The question is to express (1- i tanx) / (1+ i tanx) in polar form.
First, multiple the whole fraction by cosx. It becomes (cosx - i sin x) / (cosx + i sin x). We can find the modulus and argument easily by using the fact that "if z1=r cis a, z2=r cis b , then z1 / z2 = r1/r2 [cos(a-b) + i sin (a-b)] ". They are 1 and -2x respectively.

However, there exists another way to handle it. By multipling the whole fraction by (cosx - i sin x) / (cosx - i sin x), we can obtain (cos2x-sin2x-2cosxsinx i). The modulus is, of course, found to be 1. However, problem arose when I wanted to find the argument. Letting B be the argument, I set up "tanB = (-2cosxsinx) / (cos2x-sin2x). The equation can be written as tanB=-tan2x. Finally, both -2x and 180o-2x are found as the solutions. However, there should be two arguments for the same "polar form", right? I wonder what is wrong with that.

hi abcd8989!
abcd8989 said:
(cosx - i sin x) / (cosx + i sin x)

erm

Euler's equation ?

abcd8989 said:
The question is to express (1- i tanx) / (1+ i tanx) in polar form.
First, multiple the whole fraction by cosx. It becomes (cosx - i sin x) / (cosx + i sin x). We can find the modulus and argument easily by using the fact that "if z1=r cis a, z2=r cis b , then z1 / z2 = r1/r2 [cos(a-b) + i sin (a-b)] ". They are 1 and -2x respectively.

However, there exists another way to handle it. By multipling the whole fraction by (cosx - i sin x) / (cosx - i sin x), we can obtain (cos2x-sin2x-2cosxsinx i). The modulus is, of course, found to be 1. However, problem arose when I wanted to find the argument. Letting B be the argument, I set up "tanB = (-2cosxsinx) / (cos2x-sin2x). The equation can be written as tanB=-tan2x. Finally, both -2x and 180o-2x are found as the solutions. However, there should be two arguments for the same "polar form", right? I wonder what is wrong with that.
When you use $tan(\theta)= a/b$ for individual a and b, you lose track of the signs of a and b separately. That is, if a/b is positive, it may be that a and b are both positive (first quadrant) or it may be that a and b are both negative (third quadrant). Similarly, a/b is negative, it may be that a is negative and b positive (second quadrant) or that a is positive and b negative (fourth quadrant).

Simpler, as tiny-tim suggests, is to write
$$\frac{cos(x)- i sin(x)}{cos(x)+ i sin(x)}= \frac{e^{-ix}}{e^{ix}}= e^{-2ix}$$
to see that the modulus is 1, as you say, and the argument is -2x.

## 1. What are complex numbers?

Complex numbers are numbers that contain both a real part and an imaginary part. They are expressed in the form a + bi, where a is the real part and bi is the imaginary part (i is the imaginary unit, which is equal to the square root of -1).

## 2. How do you express a complex number in polar form?

A complex number can be expressed in polar form as r(cosθ + i sinθ), where r is the distance from the origin (also known as the modulus or absolute value) and θ is the angle formed between the positive real axis and the vector representing the complex number.

## 3. What is the significance of expressing a complex number in polar form?

Expressing a complex number in polar form allows for easier multiplication and division of complex numbers, as well as a better understanding of their geometric representation on the complex plane.

## 4. How do you convert a complex number from rectangular form to polar form?

To convert a complex number from rectangular form to polar form, you can use the formulas r = √(a² + b²) and θ = tan⁻¹(b/a), where a is the real part and b is the imaginary part.

## 5. Can you provide an example of expressing a complex number in polar form?

For example, if we have the complex number 3 + 4i, we can convert it to polar form by first finding the modulus: r = √(3² + 4²) = √(9 + 16) = √25 = 5. Then, we can find the angle θ by using the formula θ = tan⁻¹(4/3) ≈ 53.13°. Therefore, the complex number 3 + 4i can be expressed as 5(cos53.13° + i sin53.13°) in polar form.