# DeMoivre's Theorem express (sqrt(2)/2 + sqrt(2)/2 i)^8 in a+bi form

• MHB
• Elissa89
In summary, the conversation discusses how to express (sqrt(2)/2 + sqrt(2)/2 i)^8 in a+bi form using the theorem. The speaker initially tries 1(cos (2pi) +i*sin (2pi)) but is told this is incorrect. After attempting "0+1i" and "i", it is pointed out that $\sin(2\pi)$ should be evaluated as $0$.
Elissa89
So the question is:

express (sqrt(2)/2 + sqrt(2)/2 i)^8 in a+bi form

I know r=1 and tangent=pi/4

Using the theorem i get 1(cos (2pi) +i*sin (2pi)) which becomes 1(1*i)=1*i however WebAssign says this is incorrect. I've also tried "0+1i" and just "i"

What am I doing wrong?

Elissa89 said:
So the question is:

express (sqrt(2)/2 + sqrt(2)/2 i)^8 in a+bi form

I know r=1 and tangent=pi/4

Using the theorem i get 1(cos (2pi) +i*sin (2pi)) which becomes 1(1*i)=1*i however WebAssign says this is incorrect. I've also tried "0+1i" and just "i"

What am I doing wrong?

Isn't $\sin(2\pi)$ equal to $0$ instead of $1$?

You are evaluating sine and cosine incorrectly!

$cos(2\pi)+ i sin(2\pi)= 1+ i(0)= 1$.

## 1. What is DeMoivre's Theorem?

DeMoivre's Theorem is a mathematical theorem that relates trigonometric functions to complex numbers. It states that for any complex number z and any positive integer n, the nth power of z can be expressed in terms of its modulus (absolute value) and argument (angle) using trigonometric functions.

## 2. How do you apply DeMoivre's Theorem to express (sqrt(2)/2 + sqrt(2)/2 i)^8 in a+bi form?

To apply DeMoivre's Theorem, we first need to convert the given complex number to polar form. In this case, we can rewrite sqrt(2)/2 + sqrt(2)/2 i as sqrt(2) * (cos(pi/4) + i*sin(pi/4)). Then, we can use the formula z^n = r^n * (cos(n*theta) + i*sin(n*theta)), where r is the modulus and theta is the argument, to express (sqrt(2)/2 + sqrt(2)/2 i)^8 in a+bi form.

## 3. What is the modulus and argument of sqrt(2)/2 + sqrt(2)/2 i?

The modulus of sqrt(2)/2 + sqrt(2)/2 i is sqrt(2), which can be found by taking the absolute value of the complex number. The argument is pi/4, which can be found by taking the inverse tangent of the imaginary part divided by the real part.

## 4. Can DeMoivre's Theorem be used for any complex number?

Yes, DeMoivre's Theorem can be used for any complex number, as long as it is raised to a positive integer power.

## 5. Are there any other applications of DeMoivre's Theorem?

Yes, DeMoivre's Theorem has many applications in mathematics, physics, and engineering. It is commonly used to simplify complex calculations involving trigonometric functions and to solve equations involving complex numbers.

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