Find the real part of the complex number $z$ on the unit circle given by $z = e^{i heta}$ and $ heta = frac{pi}{4}$.

We are given the complex number $z$ on the unit circle:

$z = e^{i\theta}$

For $\theta = \frac{\pi}{4}$, we have:

$z = e^{i\frac{\pi}{4}}$

By Euler’s formula, $e^{i\theta} = \cos \theta + i \sin \theta$, so:

$e^{i\frac{\pi}{4}} = \cos \frac{\pi}{4} + i \sin \frac{\pi}{4}$

We know $\cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, thus:

$e^{i\frac{\pi}{4}} = \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}$

Therefore, the real part of $z$ is:

$\boxed{\frac{\sqrt{2}}{2}}$

The complex number $z$ on the unit circle is given by:

$z = e^{i heta}$

For $ heta = frac{pi}{4}$, this becomes:

$z = e^{ifrac{pi}{4}}$

Using Euler’s formula:

$e^{i heta} = cos heta + i sin heta$

Substituting $ heta = frac{pi}{4}$:

$e^{ifrac{pi}{4}} = cos frac{pi}{4} + i sin frac{pi}{4}$

Since $cos frac{pi}{4} = sin frac{pi}{4} = frac{sqrt{2}}{2}$, we get:

$e^{ifrac{pi}{4}} = frac{sqrt{2}}{2} + i frac{sqrt{2}}{2}$

Thus, the real part of $z$ is:

$oxed{frac{sqrt{2}}{2}}$

Given $z = e^{i heta}$ with $ heta = frac{pi}{4}$:

$z = e^{ifrac{pi}{4}}$

By Euler’s formula:

$e^{ifrac{pi}{4}} = cos frac{pi}{4} + i sin frac{pi}{4}$

We have $cos frac{pi}{4} = frac{sqrt{2}}{2}$:

$e^{ifrac{pi}{4}} = frac{sqrt{2}}{2} + i frac{sqrt{2}}{2}$

Thus, the real part of $z$ is:

$oxed{frac{sqrt{2}}{2}}$

Start Using PopAi Today