Question: The polar coordinates of a point are given. Find the rectangular coordinates of this point. \[…

The polar coordinates of a point are given. Find the rectangular coordinates of this point.

\[ \left( -4, \frac{\pi}{4} \right) \]

Solution

To convert polar coordinates \((-4, \frac{\pi}{4})\) to rectangular coordinates \((x, y)\), we use the formulas: \[ \begin{cases} x = r \cdot \cos(\theta) & \\ y = r \cdot \sin(\theta) & \end{cases} \] First, substitute \(r = -4\) and \(\theta = \frac{\pi}{4}\). Calculate \(x\): \[ x = -4 \cdot \cos\left(\frac{\pi}{4}\right) \] The value of \(\cos\left(\frac{\pi}{4}\right)\) is \(\frac{\sqrt{2}}{2}\). \[ x = -4 \cdot \frac{\sqrt{2}}{2} = -2\sqrt{2} \] Calculate \(y\): \[ y = -4 \cdot \sin\left(\frac{\pi}{4}\right) \] The value of \(\sin\left(\frac{\pi}{4}\right)\) is \(\frac{\sqrt{2}}{2}\). \[ y = -4 \cdot \frac{\sqrt{2}}{2} = -2\sqrt{2} \] So, the rectangular coordinates are \((-2\sqrt{2}, -2\sqrt{2})\).