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To convert rectangular coordinates \((x, y)\) into polar coordinates \((r, \theta)\), we use the following formulas:

1. **Magnitude \(r\):**
   \[
   r = \sqrt{x^2 + y^2}
   \]
2. **Angle \(\theta\):**
   \[
   \theta = \tan^{-1}\left(\frac{y}{x}\right)
   \]
   where the angle \(\theta\) is usually measured in degrees, and you must adjust the angle based on the quadrant of the point.

Given rectangular coordinates \((5, -2)\):

### Step 1: Calculate the magnitude \(r\):
\[
r = \sqrt{5^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29}
\]

### Step 2: Calculate the angle \(\theta\):
\[
\theta = \tan^{-1}\left(\frac{-2}{5}\right)
\]
Using a calculator to find the inverse tangent:
\[
\theta \approx -21.8^\circ
\]

Since the point \((5, -2)\) is in the fourth quadrant, the negative angle means it's measured clockwise from the positive x-axis.

To express the angle as a positive angle (measured counterclockwise from the positive x-axis), we can add \(360^\circ\):
\[
\theta = 360^\circ - 21.8^\circ = 338.2^\circ
\]

### Polar Coordinates:
The polar coordinates are \((\sqrt{29}, 338.2^\circ)\).

### Correct Answer:
Option **A** is the correct answer: \((\sqrt{29}, 338.2^\circ)\).

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Would you like further clarification or details?

Here are 5 related questions to further your understanding:

1. How do we determine the quadrant based on the signs of the rectangular coordinates?
2. What is the relationship between polar and rectangular coordinates in the second quadrant?
3. How would you convert polar coordinates back to rectangular coordinates?
4. Can you explain why we add \(360^\circ\) to negative angles in polar form?
5. What is the difference between degrees and radians when expressing the angle in polar coordinates?

**Tip**: Always consider the quadrant of the point when converting rectangular to polar coordinates to correctly adjust the angle \(\theta\).

Si es que tenemos las coordenadas rectangulares (5, -2), ¿Cuál es su equivalente en polares?

Learn how to convert the rectangular coordinates (5, -2) into polar coordinates. This involves calculating the magnitude and angle, adjusting the angle based on the quadrant. Suitable for students in Grades 10-12.

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