Express $$ Z=5\left(\cos 60^{\circ}+i \sin 60^{\circ}\right) $$ in Euler's form

Question

UpStudy AI · Accepted Answer

We start with the given complex number:

$$
Z=5\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)
$$

Recall Euler's formula:

$$
e^{i\theta} = \cos\theta + i\sin\theta
$$

Here, we have $$\theta = 60^{\circ}$$. Therefore, we can write:

$$
\cos 60^{\circ}+i \sin 60^{\circ} = e^{i60^{\circ}}
$$

So the complex number becomes:

$$
Z=5e^{i60^{\circ}}
$$

If you prefer the angle in radians, remember that $$60^{\circ} = \frac{\pi}{3}$$. Thus, in radians the Euler's form is:

$$
Z=5e^{i\frac{\pi}{3}}
$$
$$ Z = 5e^{i60^\circ} $$

Express in Euler’s form