Start from ##a+bi-\frac{1}{a+bi}=i##, multiply both sides with ##a+bi## and work it out. That will allow you to find the value(s) of a and b. Then use De Moivre's formula to get ##z^6##.DiamondV said:Homework Statement
2. Homework Equations [/B]
De Moivres Theorem/ Eulers formula
The Attempt at a Solution
Honestly don't know where to go with this now. I already applied De Moivres theorem at the very end. It feels like I have to do something more with either De Moivres theorem or Eulers formula. (only 2 we have done)
Samy_A said:
On the left you have a complex number, say ##X+iY##, that is equal to 0. What can you then conclude about X and Y?DiamondV said:
Where do I go form here?
then X and Y should be equal to 0. But I don't see where is the complex number on the left? I know b-ai is one but I'm not too sure about the 2abi thing?Samy_A said:
Correct.DiamondV said:
The complex number on the left is ##a²+2abi-b²-1-ai+b=(a²-b²-1+b)+(2ab-a)i##. Your computation showed that this complex number is 0. You should be able to get the value(s) of a and b from that.DiamondV said:
Ah. I see now. the entire left side was a complex number. Is this correct now?Samy_A said:
b is correct.DiamondV said:Ah. I see now. the entire left side was a complex number. Is this correct now?
Samy_A said:
Any value for a will give you a true statement if b=1/2.DiamondV said:
Correct.DiamondV said:
DiamondV said:Homework Statement
2. Homework Equations [/B]
De Moivres Theorem/ Eulers formula
The Attempt at a Solution
Honestly don't know where to go with this now. I already applied De Moivres theorem at the very end. It feels like I have to do something more with either De Moivres theorem or Eulers formula. (only 2 we have done)
De Moivre's Theorem is a mathematical formula that relates complex numbers to trigonometric functions. It states that for any complex number z and any positive integer n, (cos θ + i sin θ)^n = cos nθ + i sin nθ. Euler's Formula is a special case of De Moivre's Theorem when n = 1, which states that e^(iθ) = cos θ + i sin θ.
De Moivre's Theorem and Euler's Formula have many applications in mathematics and physics. They are commonly used to simplify calculations involving complex numbers, and they have applications in fields such as electrical engineering, signal processing, and quantum mechanics.
Euler's Formula is a special case of De Moivre's Theorem, as it can be derived by setting n = 1 in De Moivre's Theorem. Essentially, Euler's Formula is just a simplified version of De Moivre's Theorem.
Yes, De Moivre's Theorem and Euler's Formula can be used for negative values of n. In these cases, the formula will result in the reciprocal of the positive n value. For example, if n = -3, then (cos θ + i sin θ)^-3 = (cos θ + i sin θ)^3 = cos 3θ + i sin 3θ.
De Moivre's Theorem and Euler's Formula are limited to calculations involving complex numbers. They cannot be used for real numbers or other types of numbers. Additionally, they have limitations in certain circumstances, such as when n is a non-integer value or when θ is a complex number.