Find the polar equation for the curve represented by the given Cartesian equation. $x + y = 2$
A) $r = 2 ( \cos \theta + \sin \theta )$
B) $r = \frac { 1 } { \cos \theta - \sin \theta }$
C) $r = \frac { 2 } { \cos \theta + \sin \theta }$
D) $r = \frac { 2 } { \cos \theta - \sin \theta }$
E) $r = 1 ( \cos \theta + \sin \theta )$

Question

Quizplus · Accepted Answer

To convert from Cartesian to polar coordinates, we use the relationships $x=r\cos	heta$ and $y=r\sin	heta$, so we substitute those into the given equation to get
$$r\cos	heta+r\sin	heta=2$$
We can simplify this by factoring out an $r$ and using the trig identity $\cos	heta+\sin	heta=\sqrt{2}\sin\left(	heta+\frac{\pi}{4}ight)$:
$$r(\cos	heta+\sin	heta)=2 \quad\Rightarrow\quad r=\frac{2}{\cos	heta+\sin	heta}=\frac{2}{\sqrt{2}\sin\left(	heta+\frac{\pi}{4}ight)}=\frac{2}{\sin\left(	heta+\frac{\pi}{4}ight)}$$
Next, we use the double angle formula for $\sin$:
$$r=\frac{2}{2\sin\left(\frac{	heta}{2}+\frac{\pi}{8}ight)\cos\left(\frac{	heta}{2}-\frac{\pi}{8}ight)}=\frac{1}{\frac{\sin\left(\frac{	heta}{2}+\frac{\pi}{8}ight)\cos\left(\frac{	heta}{2}-\frac{\pi}{8}ight)}{\frac{1}{2}}}$$
Finally, we recognize that the expression in the denominator is the formula for the sine of the difference of angles, so we can simplify it:
$$r=\frac{1}{\sin\frac{	heta+\frac{\pi}{4}}{2}\sin\frac{\frac{\pi}{4}-\frac{	heta}{2}}{2}}=\frac{2}{\cos	heta+\sin	heta}$$
Therefore, the polar equation is $r=\frac{2}{\cos	heta+\sin	heta}$, which matches choice (C).

Find the Polar Equation for the Curve Represented by the Given