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To convert the given rectangular coordinates \((5, 5)\) to polar coordinates, we use the following formulas:

1. **Radius (r):**
   \[
   r = \sqrt{x^2 + y^2}
   \]
   where \(x = 5\) and \(y = 5\).

2. **Angle (θ):**
   \[
   \theta = \tan^{-1}\left(\frac{y}{x}\right)
   \]
   where \(x = 5\) and \(y = 5\), and we want the angle in degrees.

### Step 1: Calculate the radius \(r\)
\[
r = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} \approx 7.071
\]

### Step 2: Calculate the angle \(θ\)
\[
\theta = \tan^{-1}\left(\frac{5}{5}\right) = \tan^{-1}(1)
\]
From a standard trigonometric table, we know:
\[
\theta = 45^\circ
\]

### Final Polar Coordinates:
The polar coordinates are \((r, \theta) = (7.071, 45^\circ)\).

Would you like further details on the steps or have any questions? Here are 5 related questions to explore further:

1. How would the polar coordinates change if the point was \((-5, 5)\)?
2. How do we convert polar coordinates back to rectangular coordinates?
3. What would the polar coordinates be if the rectangular point was on the y-axis, like \((0, 5)\)?
4. How does the tangent inverse function help in finding the angle in polar coordinates?
5. What is the significance of the radius \(r\) in polar coordinates?

**Tip:** Always pay attention to the quadrant of the point to ensure the correct angle is used for the polar coordinates.

Find polar coordinates of the point that has rectangular coordinates (5, 5).

Learn how to convert the rectangular coordinates (5, 5) to polar coordinates. This involves using key trigonometric formulas for the radius and angle, applying the Pythagorean Theorem, and understanding the tangent inverse function. Suitable for students in Grades 9-12.

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