Convert the point from Cartesian to polar coordinates. Write your answer in radians. Round to the nearest hundredth.

(5,5)


To convert the Cartesian coordinates \((5, 5)\) to polar coordinates, we follow these steps:

1. **Calculate the radius \(r\)**:
   \[
   r = \sqrt{x^2 + y^2}
   \]
   Substituting the given coordinates \((5, 5)\):
   \[
   r = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \approx 7.07
   \]

2. **Calculate the angle \(\theta\)** in radians:
   \[
   \theta = \tan^{-1}\left(\frac{y}{x}\right)
   \]
   Substituting the given coordinates \((5, 5)\):
   \[
   \theta = \tan^{-1}\left(\frac{5}{5}\right) = \tan^{-1}(1) = \frac{\pi}{4} \approx 0.79 \text{ radians}
   \]

Therefore, the polar coordinates of the point \((5, 5)\) are approximately:
\[
(r, \theta) = (7.07, 0.79)
\]

Would you like more details or have any questions? Here are some related questions:

1. What are the formulas for converting Cartesian coordinates to polar coordinates?
2. How do you find the radius \(r\) in polar coordinates?
3. How do you find the angle \(\theta\) in polar coordinates?
4. Why is the angle \(\theta\) expressed in radians?
5. Can you convert polar coordinates back to Cartesian coordinates?
6. What is the significance of the tangent function in finding \(\theta\)?
7. How do you handle coordinates in different quadrants when converting to polar form?
8. What are some practical applications of polar coordinates?

**Tip:** When converting to polar coordinates, always consider the quadrant of the original Cartesian coordinates to determine the correct angle \(\theta\).

Learn how to convert Cartesian coordinates (5, 5) to polar coordinates using the distance formula and trigonometric functions. The radius is approximately 7.07 and the angle is approximately 0.79 radians. Suitable for high school students.

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