Find a Cartesian equation for the line whose polar equation is given.
-$r \cos \left( \theta - \frac { \pi } { 6 } \right) = 2$
A) $y = 4 - \frac { \sqrt { 3 } } { 3 } x$
B) $y = \sqrt { 3 } x + 2$
C) $y = 4 - \sqrt { 3 } x$
D) $y = 2 - \sqrt { 2 } x$

Question

Quizplus · Accepted Answer

Using the trigonometric identity $\cos(	heta-\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\cos 	heta + \frac{1}{2}\sin 	heta$, we can rewrite the polar equation as:

$$r\left(\frac{\sqrt{3}}{2}\cos 	heta + \frac{1}{2}\sin 	hetaight) = 2$$

We can then convert this to a Cartesian equation using the conversions $x = r\cos 	heta$ and $y = r\sin 	heta$:

$$\frac{\sqrt{3}}{2}x + \frac{1}{2}y = 2$$

Simplifying by multiplying both sides by 2, we get:

$$\sqrt{3}x + y = 4$$

This is in the form $y = mx + b$, where $m = \sqrt{3}$ and $b = 4$. Therefore, the equation is a line with slope $\sqrt{3}$ and $y$-intercept 4. The correct choice is C.

Find a Cartesian Equation for the Line Whose Polar Equation rcos⁡(θ−π6)=2r \cos \left( \theta - \frac { \pi } { 6 } \right) = 2rcos(θ−6π​)=2

Find a Cartesian Equation for the Line Whose Polar Equation $r \cos \left( \theta - \frac { \pi } { 6 } \right) = 2$