To convert polar coordinates $(r, \theta)$ to rectangular coordinates $(x, y)$, we use the formulas:
$$x = r \cos \theta$$
$$y = r \sin \theta$$
Here is the step-by-step conversion for each point:
A. $(5, \pi)$
* $r = 5$, $\theta = \pi$
* $x = 5 \cos(\pi) = 5(-1) = -5$
* $y = 5 \sin(\pi) = 5(0) = 0$
*
Rectangular Coordinate: $(-5, 0)$
B. $\left(8, \frac{5\pi}{6}\right)$
* $r = 8$, $\theta = \frac{5\pi}{6}$
* The reference angle is $\frac{\pi}{6}$. In the second quadrant, cosine is negative and sine is positive.
* $\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$
* $\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}$
* $x = 8 \left(-\frac{\sqrt{3}}{2}\right) = -4\sqrt{3}$
* $y = 8 \left(\frac{1}{2}\right) = 4$
*
Rectangular Coordinate: $(-4\sqrt{3}, 4)$
C. $\left(4, -\frac{\pi}{3}\right)$
* $r = 4$, $\theta = -\frac{\pi}{3}$
* This angle is in the fourth quadrant. Cosine is positive, sine is negative.
* $\cos\left(-\frac{\pi}{3}\right) = \frac{1}{2}$
* $\sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}$
* $x = 4 \left(\frac{1}{2}\right) = 2$
* $y = 4 \left(-\frac{\sqrt{3}}{2}\right) = -2\sqrt{3}$
*
Rectangular Coordinate: $(2, -2\sqrt{3})$
D. $\left(-2, \frac{\pi}{6}\right)$
* $r = -2$, $\theta = \frac{\pi}{6}$
* Note: A negative radius means you go in the opposite direction of the angle.
* $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$
* $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$
* $x = -2 \left(\frac{\sqrt{3}}{2}\right) = -\sqrt{3}$
* $y = -2 \left(\frac{1}{2}\right) = -1$
*
Rectangular Coordinate: $(-\sqrt{3}, -1)$
E. $\left(-3, -\frac{\pi}{4}\right)$
* $r = -3$, $\theta = -\frac{\pi}{4}$
* $\cos\left(-\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$
* $\sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}$
* $x = -3 \left(\frac{\sqrt{2}}{2}\right) = -\frac{3\sqrt{2}}{2}$
* $y = -3 \left(-\frac{\sqrt{2}}{2}\right) = \frac{3\sqrt{2}}{2}$
*
Rectangular Coordinate: $\left(-\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}\right)$
Final Answer:
A. $(-5, 0)$
B. $(-4\sqrt{3}, 4)$
C. $(2, -2\sqrt{3})$
D. $(-\sqrt{3}, -1)$
E. $\left(-\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}\right)$
Parent Tip: Review the logic above to help your child master the concept of polar coordinates worksheet.
