Questions: The rectangular coordinates of a point are given. Find polar coordinates for the point. (3.4,-3.7)

Solution Steps

To convert rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\), we use the formulas: \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan\left(\frac{y}{x}\right)\). The angle \(\theta\) should be adjusted based on the quadrant in which the point lies.

To find the radius \( r \) of the point in polar coordinates, we use the formula: \[ r = \sqrt{x^2 + y^2} \] Substituting the values \( x = 3.4 \) and \( y = -3.7 \): \[ r = \sqrt{(3.4)^2 + (-3.7)^2} = \sqrt{11.56 + 13.69} = \sqrt{25.25} \approx 5.0249 \]

To find the angle \( \theta \), we use the formula: \[ \theta = \arctan\left(\frac{y}{x}\right) \] Substituting the values: \[ \theta = \arctan\left(\frac{-3.7}{3.4}\right) \approx -0.8276 \text{ radians} \] To convert this angle to degrees: \[ \theta_{\text{degrees}} = -47.4195^\circ \]

Final Answer