Question: Convert the following polar points to Cartesian coordinates. (a) \((5, \frac{5\pi}{4})\) (b)…

Convert the following polar points to Cartesian coordinates.

(a) \((5, \frac{5\pi}{4})\)

(b) \((-2, \frac{3\pi}{4})\)

Solution

To convert polar coordinates \((r, \theta)\) to Cartesian coordinates \((x, y)\), we use the formulas: \[ x = r \cdot \cos(\theta) \] \[ y = r \cdot \sin(\theta) \] Let’s solve the problems: (a) \((5, \frac{5\pi}{4})\) First, calculate \(x\): \[ x = 5 \cdot \cos\left(\frac{5\pi}{4}\right) \] Since \(\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}\), \[ x = 5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = -\frac{5\sqrt{2}}{2} \] Now, calculate \(y\): \[ y = 5 \cdot \sin\left(\frac{5\pi}{4}\right) \] Since \(\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}\), \[ y = 5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = -\frac{5\sqrt{2}}{2} \] The Cartesian coordinates are: \((- \frac{5\sqrt{2}}{2}, - \frac{5\sqrt{2}}{2})\) (b) \((-2, \frac{3\pi}{4})\) First, calculate \(x\): \[ x = -2 \cdot \cos\left(\frac{3\pi}{4}\right) \] Since \(\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}\), \[ x = -2 \cdot \left(-\frac{\sqrt{2}}{2}\right) = \sqrt{2} \] Now, calculate \(y\): \[ y = -2 \cdot \sin\left(\frac{3\pi}{4}\right) \] Since \(\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}\), \[ y = -2 \cdot \left(\frac{\sqrt{2}}{2}\right) = -\sqrt{2} \] The Cartesian coordinates are: \((\sqrt{2}, -\sqrt{2})\)