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

Question

Quizplus · Accepted Answer

Using the double angle identity for cosine, we have: 
$$r\cos\left(\theta+\frac{4\pi}{3}\right) = 2\sqrt{3}$$
$$r\cos\theta\cos\frac{4\pi}{3} - r\sin\theta\sin\frac{4\pi}{3} = 2\sqrt{3}$$
$$-\frac{1}{2}r\cos\theta - \frac{\sqrt{3}}{2}r\sin\theta = 2\sqrt{3}$$
Multiplying both sides by $-2$ and simplifying, we get:
$$r\sqrt{3} + 3x = -8$$
Substituting $r=\sqrt{x^2+y^2}$, we get:
$$\sqrt{x^2+y^2}\sqrt{3} + 3x = -8$$
Simplifying and rearranging, we get:
$$\boxed{y=\frac{\sqrt{3}}{3}x+4}$$

Find a Cartesian Equation for the Line Whose Polar Equation rcos⁡(θ+4π3)=23r \cos \left( \theta + \frac { 4 \pi } { 3 } \right) = 2 \sqrt { 3 }rcos(θ+34π​)=23​

Find a Cartesian Equation for the Line Whose Polar Equation $r \cos \left( \theta + \frac { 4 \pi } { 3 } \right) = 2 \sqrt { 3 }$