Find the polar coordinates of a point with Cartesian coordinates (*x*,*y*)=(93√2,92).

To find the polar coordinates \((r, \theta)\) of a point with given Cartesian coordinates \((x, y) = (93\sqrt{2}, 92)\), we use the following formulas:

1. **Magnitude (radius) \( r \):**
\[
r = \sqrt{x^2 + y^2}
\]
2. **Angle \( \theta \):**
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\]

### Step 1: Calculate \( r \)

Given \( x = 93\sqrt{2} \) and \( y = 92 \):

\[
r = \sqrt{(93\sqrt{2})^2 + 92^2}
\]
\[
r = \sqrt{(93^2 \cdot 2) + 92^2}
\]
\[
r = \sqrt{(8649 \cdot 2) + 8464}
\]
\[
r = \sqrt{17298 + 8464}
\]
\[
r = \sqrt{25762}
\]
\[
r \approx 160.50
\]

### Step 2: Calculate \( \theta \)

\[
\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{92}{93\sqrt{2}}\right)
\]

To simplify the expression inside the arctan function:

\[
\theta = \tan^{-1}\left(\frac{92}{93\sqrt{2}}\right) \approx \tan^{-1}(0.700)
\]

Using a calculator:

\[
\theta \approx 35.00^\circ
\]

### Final Answer
The polar coordinates of the point \((93\sqrt{2}, 92)\) are approximately:
\[
\left( r, \theta \right) \approx (160.50, 35.00^\circ)
\]

Would you like further details or have any other questions?

### Related Questions
1. How do you convert polar coordinates back to Cartesian coordinates?
2. What is the general formula for converting Cartesian coordinates to polar coordinates?
3. Can polar coordinates be negative, and if so, what does that represent?
4. How do you find the angle \( \theta \) when \( x \) is negative?
5. How can polar coordinates be visualized on the polar grid?
6. What is the relationship between polar and Cartesian coordinate systems?
7. How do you handle polar coordinates when \( y = 0 \) or \( x = 0 \)?
8. How do trigonometric identities help in converting between coordinate systems?

**Tip:** In polar coordinates, the angle \( \theta \) is usually measured counterclockwise from the positive x-axis, and the radius \( r \) is always non-negative.

Learn how to find polar coordinates from Cartesian coordinates (93√2, 92) using trigonometric identities and coordinate system conversions. This problem involves calculating the magnitude (radius) and angle of a point in the polar coordinate system. Suitable for students in Grades 10-12.

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