Find approximations for the coordinates of the point $(x, y)$ at the given angle $\theta$ on a circle of radius $r=4.25$ centered at the origin. Round your answers to two decimal places. (a) $\theta=501^{\circ}$

Step 1 ：Given a circle of radius \(r = 4.25\) centered at the origin, we are asked to find the coordinates of the point at the angle \(\theta = 501^\circ\).

Step 2 ：First, we need to convert the angle from degrees to radians. The conversion factor is \(\frac{\pi}{180}\).

Step 3 ：Since a circle is 360 degrees, we can subtract 360 from the angle until it is less than 360 to simplify the calculation. So, \(\theta = 501^\circ - 360^\circ = 141^\circ\).

Step 4 ：Convert \(\theta\) to radians: \(\theta = 141^\circ \times \frac{\pi}{180} = 2.4609142453120048\) radians.

Step 5 ：The coordinates of a point on a circle centered at the origin can be found using the formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\), where \(r\) is the radius of the circle and \(\theta\) is the angle in radians.

Step 6 ：Calculate the x-coordinate: \(x = 4.25 \cos(2.4609142453120048) = -3.302870336192126\).

Step 7 ：Calculate the y-coordinate: \(y = 4.25 \sin(2.4609142453120048) = 2.674611661961809\).

Step 8 ：Round the x and y coordinates to two decimal places to get the final answer: \(x = -3.30\), \(y = 2.67\).

Step 9 ：Final Answer: The coordinates of the point are approximately \(\boxed{(-3.30, 2.67)}\).