To solve this problem of finding the polar coordinates of the given point $(2, 2\sqrt{3})$, we follow these steps:

Step 1: Find $r$, the radial distance

The radial distance $r$ is calculated as: $r = \sqrt{x^2 + y^2}$ Here: $x = 2, \quad y = 2\sqrt{3}$ Substitute these values: $r = \sqrt{(2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4$

Step 2: Find $\theta$, the angle

The angle $\theta$ is determined using the formula: $\theta = \tan^{-1}\left(\frac{y}{x}\right)$ Substitute $x = 2$ and $y = 2\sqrt{3}$: $\theta = \tan^{-1}\left(\frac{2\sqrt{3}}{2}\right) = \tan^{-1}(\sqrt{3})$

From trigonometric knowledge: $\tan^{-1}(\sqrt{3}) = 60^\circ$

Step 3: Convert $\theta$ to radians

To convert degrees to radians, use: $\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$ So: $\theta = 60^\circ \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}$

Final Answer:

The polar coordinates of the point are: $(4, 60^\circ) \quad \text{or} \quad (4, \frac{\pi}{3} \text{ radians})$

Would you like more details or have further questions?

Here are 5 related questions for deeper understanding:

1. How do you derive the formula for converting Cartesian coordinates to polar coordinates?
2. Why do we use $r > 0$ for polar coordinates?
3. Can there be multiple angles that represent the same point in polar coordinates? Why?
4. How would the calculations change if the point was in a different quadrant?
5. What is the importance of simplifying trigonometric values like $\tan^{-1}(\sqrt{3})$?

Tip: Always remember to check which quadrant your point is in before determining the angle $\theta$.