# Complex Numbers identity help

• lunds002
In summary, to express (z1/z2)^3 in the form z = x + yi, we can use the identity eiθ=cosθ+isinθ to simplify the calculations. The final answer is z=(√2)a3/2b3 - (√2)a3/2b3 i.
lunds002

## Homework Statement

Let z1 = a (cos (pi/4) + i sin (pi/4) ) and z2 = b (cos (pi/3) + i sin (pi/3))
Express (z1/z2)^3 in the form z = x + yi.

]2. Homework Equations [/b]

## The Attempt at a Solution

a(cos (pi/4) + i sin (pi/4))
b (cos (pi/3) + i sin (pi/3))

I then multiplied the top and bottom by the bottom reciprocal : cos (pi/3) - i sin (pi/3) to get

a(cos (pi/4) x cos (pi/3) + sin (pi/4) sin (pi/3)) + i (sin (pi/4) x cos (pi/3) - sin (pi/3) x cos (pi/4)

Do you know the identity e=cosθ+isinθ?

It will make your calculations easier since you can express it in terms of the exponential function.

right, i forget that. thank you

I'm not sure if I'm right, I am stuck on this question as well.

I got z=(√2)a3/2b3 - (√2)a3/2b3 i

That's right.

vela said:
That's right.

That's great. I'm still kind of new to this topic, still not confident with it and I have an unit test coming up in less than a week's time!

## 1. What is the definition of a complex number?

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. The value of i is defined as the square root of -1.

## 2. How do you add and subtract complex numbers?

To add or subtract complex numbers, you simply combine the real parts (a) and the imaginary parts (bi) separately. For example, (3+2i) + (5+4i) = (3+5) + (2+4)i = 8 + 6i.

## 3. What is the purpose of the complex numbers identity?

The complex numbers identity, also known as Euler's formula, is used to represent complex numbers in terms of trigonometric functions. It is also helpful in solving complex equations and simplifying complex expressions.

## 4. How do you multiply and divide complex numbers?

To multiply complex numbers, you use the FOIL method, just like with binomials. For example, (3+2i)(5+4i) = 3(5) + 3(4i) + 2i(5) + 2i(4i) = 15 + 12i + 10i + 8i^2 = 15 + 22i - 8 = 7 + 22i. To divide complex numbers, you use the fact that the product of a number and its conjugate is a real number. For example, (3+2i)/(5+4i) = (3+2i)(5-4i)/(5+4i)(5-4i) = (15-12i+10i-8i^2)/(25-20i+20i-16i^2) = (23+2i)/41.

## 5. How are complex numbers used in real life?

Complex numbers are used in various fields of science and engineering, such as electrical engineering, signal processing, and quantum mechanics. They are also used in finance, for example in calculating interest rates and modeling stock market fluctuations. In physics, complex numbers are used to represent the amplitude and phase of electromagnetic waves. They also have applications in computer graphics and cryptography.

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