Questions: Express the Cartesian coordinates (5,5) in polar coordinates in at least two different ways. Write the point in polar coordinates with an angle in the range (0 leq theta<2 pi). Type an ordered pair. Type an exact answer, using (pi) as needed.

Solution Steps

To convert Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\), we first calculate the radius \(r\) using the Pythagorean theorem: \[ r = \sqrt{x^2 + y^2} \] Given \(x = 5\) and \(y = 5\): \[ r = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \approx 7.071 \]

Next, we calculate the angle \(\theta\) using the arctangent function: \[ \theta = \arctan\left(\frac{y}{x}\right) \] Given \(x = 5\) and \(y = 5\): \[ \theta = \arctan\left(\frac{5}{5}\right) = \arctan(1) = \frac{\pi}{4} \approx 0.7854 \]

The calculated angle \(\theta = \frac{\pi}{4}\) is already in the range \([0, 2\pi)\). Therefore, no adjustment is needed.

To provide a second representation of the polar coordinates, we can add \(2\pi\) to the angle: \[ \theta_2 = \theta + 2\pi = \frac{\pi}{4} + 2\pi = \frac{9\pi}{4} \approx 7.0686 \]

Final Answer