Questions: The polar coordinates of a point are (r=5.90 ,m) and (theta=230^circ). What are the Cartesian coordinates of this point? x= m y= m

Solution Steps

To convert polar coordinates to Cartesian coordinates, we use the formulas \( x = r \cdot \cos(\theta) \) and \( y = r \cdot \sin(\theta) \). Here, \( r \) is the radial distance and \( \theta \) is the angle in degrees. We first convert the angle from degrees to radians since Python's trigonometric functions use radians. Then, we calculate the \( x \) and \( y \) values using the given formulas.

To convert the angle from degrees to radians, we use the formula: \[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \] Given \(\theta_{\text{degrees}} = 230\), we find: \[ \theta_{\text{radians}} = 230 \times \frac{\pi}{180} \approx 4.0143 \]

Using the formulas for converting polar coordinates \((r, \theta)\) to Cartesian coordinates \((x, y)\): \[ x = r \cdot \cos(\theta_{\text{radians}}) \] \[ y = r \cdot \sin(\theta_{\text{radians}}) \] Substituting the given values \(r = 5.90\) and \(\theta_{\text{radians}} \approx 4.0143\): \[ x = 5.90 \cdot \cos(4.0143) \approx -3.792 \] \[ y = 5.90 \cdot \sin(4.0143) \approx -4.520 \]

Final Answer