223. Write the complex number $z=1+i$ in trigonometric form.
Solution. $1^{0}$. Since $a=1, b=1$, then $r=|z|=\sqrt{1^{2}+1^{2}}=\sqrt{2}$. $2^{0}$. Let's represent the number $z$ geometrically (Fig. 10). We see that the number $z$ corresponds to the point $Z$, lying in the first quadrant, and the vector $\vec{z}$.

$3^{6}$. We form the ratios $\cos \varphi=a / r$ and $\sin \varphi=b / r$, i.e.,

$$
\cos \varphi=1 / \sqrt{2}=\sqrt{2} / 2, \sin \varphi=1 / \sqrt{2}=\sqrt{2} / 2
$$

These relations correspond to an angle $\varphi=45^{\circ}$ or $\varphi=\pi / 4$ in the first quadrant.

$4^{0}$. Since $r=\sqrt{2}, \varphi=45^{\circ}$ or $\varphi=\pi / 4$, the trigonometric form of the given complex number is

$z=1+i=\sqrt{2}\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)$ or $z=1+i=\sqrt{2}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$.

![](https://cdn.mathpix.com/cropped/2024_05_22_db4d450c77c65a914ec1g-103.jpg?height=304&width=325&top_left_y=244&top_left_x=168)

Fig. 10

![](https://cdn.mathpix.com/cropped/2024_05_22_db4d450c77c65a914ec1g-103.jpg?height=375&width=417&top_left_y=170&top_left_x=645)

Fig. 11