2.3. The Cartesian coordinates of point $A(4,3)$ are known. Find the coordinates of this point in the polar coordinate system, if the pole is located at the origin of the Cartesian coordinate system, and the polar axis coincides in direction with the $O X$ axis.
Solution. Let's use the formulas connecting polar coordinates with Cartesian ones:

$$
\begin{gathered}
\rho=\sqrt{x^{2}+y^{2}}=\sqrt{4^{2}+3^{2}}=\sqrt{25}=5 \\
\tan \varphi=3 / 4 ; \varphi=\arctan 0.75 \approx 36^{\circ} 53'
\end{gathered}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_f5a9bb7960904462604bg-095.jpg?height=280&width=356&top_left_y=1190&top_left_x=146)

Fig. 2.10

It is recommended to independently

| $\varphi$ | 0 | $\pi / 4$ | $\pi / 2$ | $3 \pi / 4$ | $\pi$ | $5 \pi / 4$ | $3 \pi / 2$ | $7 \pi / 4$ | $2 \pi$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| p | 0 | $\pi / 8 \approx$  $\approx 0.39$ | $\pi / 4 \approx$  $\approx 0.79$ |  | $\pi / 2 \approx$  $\approx 1.57$ | $5 \pi / 8 \approx$  $\approx 1.96$ | $3 \pi / 4=$  $=2.11$ | $7 \pi / 8 \approx$  $\approx 2.75$ | $\pi \approx$  $\approx 3.14$ |

make a drawing for the problem.

Answer: $\rho=5 ; \varphi=\arctan 0.75 \approx$ $\approx 36^{\circ} 53'$.