Converting fifth roots from polar form to complex

In summary, The conversation is about finding the fifth roots of unity, specifically in the form of cos(2kpi/5) + isin(2kpi/5). The question was how to convert the rest of the roots to complex numbers, and the suggestion was to use common triangles like 45-45-90 and 30-60-90. However, there is no known triangle for 2pi/5. The response was that the given form is already a complex number and can be converted to polar form. The questioner then clarified that they wanted to convert it into a+bi form, which was already done by assigning a to cos(2kpi/5) and b to sin(2kpi
  • #1
Braka
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Homework Statement


I am looking for the fifth roots of unity, which I believe come in the form of:

cos(2kpi/5) + isin(2kpi/5), k=1,2,3,4,5 and when k=5, the complex number is 1.

how do you convert the rest to complex numbers? Normally, I use common triangles like:

45-45-90 and 30-60-90

but I don't think there is a triangle for 2pi/5.


Homework Equations





The Attempt at a Solution

 
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  • #2
I don't understand your question. cos(2kpi/5) + isin(2kpi/5) is a complex number. Do you want to convert that into polar form? It should be pretty obvious.
 
  • #3
Its obvious? I can't recall how to get it. What I meant is I wanted to put it in a +bi form. That is my problem. Sorry for stating it in a confusing manner
 
  • #4
It is already in the form a+bi.
a=cos(2kpi/5)
b=sin(2kpi/5)
 

1. What is the formula for converting fifth roots from polar form to complex?

The formula for converting fifth roots from polar form to complex is z = r^(1/5) * (cos((theta+2k*pi)/5) + i*sin((theta+2k*pi)/5)), where r is the magnitude or modulus of the polar form and theta is the argument or angle in radians.

2. How do you determine the number of roots when converting from polar form to complex?

The number of roots when converting from polar form to complex is equal to the degree of the root, which in this case is 5. Therefore, there will be 5 distinct roots for each polar form.

3. Can you give an example of converting a fifth root from polar form to complex?

Sure, let's say we have the polar form z = 4(cos(3pi/4) + i*sin(3pi/4)). To convert this to complex form, we use the formula z = r^(1/5) * (cos((theta+2k*pi)/5) + i*sin((theta+2k*pi)/5)), where r = 4 and theta = 3pi/4. Plugging these values in, we get z = 4^(1/5) * (cos((3pi/4+2k*pi)/5) + i*sin((3pi/4+2k*pi)/5)). Simplifying further, we get z = 4^(1/5) * (cos((3pi/20+k*pi/5) + i*sin((3pi/20+k*pi/5)), where k = 0, 1, 2, 3, 4. These are the 5 distinct fifth roots of z.

4. Why is it important to convert from polar form to complex form?

Converting from polar form to complex form allows us to easily perform operations such as addition, subtraction, multiplication, and division on complex numbers. It also helps us visualize complex numbers on the complex plane, which is useful in many mathematical and scientific applications.

5. Can you convert any polar form to complex form using the same formula?

Yes, the formula for converting fifth roots from polar form to complex can be used for any polar form. However, for other roots (e.g. square roots, cube roots), a different formula will be needed. It's important to know which formula to use for the specific root you are converting.

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