Given that $z = re^{i\theta}$, what is the value of $\frac{z}{\overline{z}}$?
Understand the Problem
The question asks us to find the value of $\frac{z}{\overline{z}}$ given that $z = re^{i\theta}$. This involves understanding complex numbers in polar form and their conjugates.
Answer
$e^{2i\theta}$
Answer for screen readers
$e^{2i\theta}$
Steps to Solve
- Represent the conjugate of z
Given $z = re^{i\theta}$, the conjugate of $z$, denoted as $\overline{z}$, is found by negating the imaginary part of $z$. In polar form, this means negating the angle $\theta$. Thus, $\overline{z} = re^{-i\theta}$.
- Divide z by its conjugate
Now, we compute $\frac{z}{\overline{z}}$:
$$ \frac{z}{\overline{z}} = \frac{re^{i\theta}}{re^{-i\theta}} $$
- Simplify the expression
We can simplify this expression by canceling the $r$ terms and using the properties of exponents:
$$ \frac{re^{i\theta}}{re^{-i\theta}} = \frac{e^{i\theta}}{e^{-i\theta}} = e^{i\theta} \cdot e^{i\theta} = e^{2i\theta} $$
Therefore, $\frac{z}{\overline{z}} = e^{2i\theta}$.
$e^{2i\theta}$
More Information
The result $e^{2i\theta}$ is also a complex number on the unit circle in the complex plane, with an angle of $2\theta$ relative to the positive real axis. Also, recall Euler's formula $e^{ix} = \cos(x) + i\sin(x)$, therefore $e^{2i\theta} = \cos(2\theta) + i\sin(2\theta)$.
Tips
A common mistake is incorrectly finding the conjugate of $z$. Remember that the conjugate $re^{i\theta}$ is $re^{-i\theta}$, not $-re^{i\theta}$. Also, be careful when applying exponent rules to simplify the expression.
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