a) \( \begin{aligned} \theta & =\frac{7 \pi}{6} \mathrm{rad} \\ r & =7,4 \mathrm{~cm}\end{aligned} \)

To convert polar coordinates \((r, \theta)\) to Cartesian coordinates \((x, y)\), you can use the formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Plugging in your values: - For \(x\): \(x = 7.4 \cos\left(\frac{7\pi}{6}\right) = 7.4 \cdot -\frac{\sqrt{3}}{2} \approx -6.4 \, \text{cm}\) - For \(y\): \(y = 7.4 \sin\left(\frac{7\pi}{6}\right) = 7.4 \cdot -\frac{1}{2} \approx -3.7 \, \text{cm}\) So, the Cartesian coordinates are approximately \((-6.4 \, \text{cm}, -3.7 \, \text{cm})\). In the context of polar coordinates, understanding the angles can lend fun insights into the unit circle! For instance, \(\frac{7\pi}{6}\) radians corresponds to an angle of \(210^\circ\), which lies in the third quadrant where both \(x\) and \(y\) coordinates are negative, just like you've computed. Instead of merely crunching numbers, think of it as placing a vector in a cool circular playground! If you're often working with angles in radians, it's handy to remember that converting between radians and degrees is just a couple of steps. To go from radians to degrees, multiply the radian measure by \(\frac{180^\circ}{\pi}\). So, for \(\frac{7\pi}{6}\), that gives you \(210^\circ\) directly. This can be really useful for visualizing where angles lie on the unit circle or in problems involving trigonometry.