Chapter 10 - Parametric Equations and Polar Coordinates - 10.3 Exercises - Page 687: 27

a. Polar equation is easier to derive. $\displaystyle \theta=\frac{\pi}{6}$ b. Cartesian equation is easier to derive. $x=3$

$a.$ A nonvertical line has a Cartesian equation $y=mx+b$ Since the line passes through the origin, b=0, and the slope m equals $\displaystyle \tan\frac{\pi}{6}=\frac{1}{\sqrt{3}}$.... so the line equation is $y=\displaystyle \frac{1}{\sqrt{3}}x.$ In polar coordinates, $\displaystyle \theta=\frac{\pi}{6}$ is the equation of the line passing through the origin, making the given angle with the x-axis. Polar equation is easier to derive. $\displaystyle \theta=\frac{\pi}{6}$ $b.$ In Cartesian coordinates, the vertical line here has an equation $x=3.$ This is as simple as it gets. In polar coordinates, lines not passing through the origin have both r and $\theta$ varying from point to point, so it would be more complicated to use a polar equation here.