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 each point: (a) \( (5, \frac{5\pi}{4}) \): First, compute \( x \): \[ x = 5 \cdot \cos\left(\frac{5\pi}{4}\right) \] Since \( \cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \), we have: \[ x = 5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = -\frac{5\sqrt{2}}{2} \] Next, compute \( y \): \[ y = 5 \cdot \sin\left(\frac{5\pi}{4}\right) \] Since \( \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \), we have: \[ y = 5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = -\frac{5\sqrt{2}}{2} \] Thus, the Cartesian coordinates are: \[ (-\frac{5\sqrt{2}}{2}, -\frac{5\sqrt{2}}{2}) \] (b) \( (-2, \frac{3\pi}{4}) \): First, compute \( x \): \[ x = -2 \cdot \cos\left(\frac{3\pi}{4}\right) \] Since \( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \), we have: \[ x = -2 \cdot \left(-\frac{\sqrt{2}}{2}\right) = \frac{2\sqrt{2}}{2} = \sqrt{2} \] Next, compute \( y \): \[ y = -2 \cdot \sin\left(\frac{3\pi}{4}\right) \] Since \( \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \), we have: \[ y = -2 \cdot \left(\frac{\sqrt{2}}{2}\right) = -\sqrt{2} \] Thus, the Cartesian coordinates are: \[ (\sqrt{2}, -\sqrt{2}) \]