Complex Numbers - Complex Roots of Unity

In summary, the conversation was about a math problem involving complex numbers and the use of half-angle formulae to solve it. The expert provided a summary of the conversation and explained how the double-angle formulae can be used to derive the half-angle formulae. The person seeking help thanked the expert for their prompt reply and understanding.
  • #1
shabi
5
0
Need help with this please:

Homework Statement


(1 + cosθ + isinθ) / (1 - cosθ - isinθ) = icotθ/2

The first step in the solutions shows:

(2cos^2θ/2 + i2sinθ/2cosθ/2) / (2sin^2θ/2 - i2sinθ/2cosθ/2)

Homework Equations



I can't get there.

The Attempt at a Solution



I tried multiplying by: (1 - cosθ + isinθ) / (1 - cosθ + isinθ), with no luck.

the top line of my attempt shows i2sinθ + i2sinθcosθ
because 1-cos^2θ=sin^2θmy signs could be wrong but still. how does the θ become θ/2. this is doing my head in. maybe just sleep on it.
 
Last edited:
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  • #2
shabi said:
Need help with this please:

Homework Statement


(1 + cosθ + isinθ) / (1 - cosθ - isinθ) = icotθ/2

The first step in the solutions shows:

(2cos^2θ/2 + i2sinθ/2cosθ/2) / (2sin^2θ/2 - i2sinθ/2cosθ/2)

Homework Equations



I can't get there.

The Attempt at a Solution



I tried multiplying by: (1 - cosθ + isinθ) / (1 - cosθ + isinθ), with no luck.

the top line of my attempt shows i2sinθ + i2sinθcosθ
because 1-cos^2θ=sin^2θmy signs could be wrong but still. how does the θ become θ/2. this is doing my head in. maybe just sleep on it.

OK, you know from the standard double-angle formulae that [itex]\sin{2\alpha} = 2\sin\alpha\cos\alpha[/itex] and that [itex]\cos{2\alpha} = 2\cos^2{\alpha} - 1 = 1 - 2\sin^2{\alpha}[/itex].

Now put [itex]\alpha = \frac{\theta}{2}[/itex].

The results you get are often called the half-angle formulae, but it's not worth remembering them specifically because they're so easily derived from the double-angle formulae.
 
  • #3
Thanks for the prompt reply and pointing that out!

So simple now i can see that.
 
  • #4
shabi said:
Thanks for the prompt reply and pointing that out!

So simple now i can see that.

You're welcome.
 

1. What are complex numbers?

Complex numbers are numbers that contain both a real and imaginary component. They are written in the form a + bi, where a is the real part and bi is the imaginary part. They are used to represent quantities that cannot be described by just real numbers.

2. What are complex roots of unity?

Complex roots of unity are solutions to the equation x^n = 1, where n is a positive integer. They are complex numbers that, when raised to the nth power, result in 1. They are important in mathematics, particularly in the study of trigonometry and number theory.

3. How do you find complex roots of unity?

The complex roots of unity can be found by using the formula e^(2πki/n), where k is an integer and n is the degree of the root. This formula determines the angle at which the root will lie on the unit circle in the complex plane. By plugging in different values of k, all n complex roots of unity can be found.

4. What is the importance of complex roots of unity?

Complex roots of unity have many applications in mathematics and physics. They can be used to solve polynomial equations, represent periodic functions, and describe the behavior of waves. They also have connections to other areas of math, such as group theory and graph theory.

5. Can complex roots of unity be expressed in other forms?

Yes, complex roots of unity can also be expressed in trigonometric form as cos(2πk/n) + i sin(2πk/n), where k is an integer and n is the degree of the root. They can also be written in exponential form as e^(2πki/n). All of these forms are equivalent and can be used to represent complex roots of unity.

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