Convert to rectangular form. $$r \sin \left( \theta + \frac { \pi } { 4 } \right) = 4$$
A) $y = - x + 8 \sqrt { 2 }$
B) $y = - x + 4 \sqrt { 2 }$
C) $y = - x \sqrt { 2 } + 4 \sqrt { 2 }$
D) $y = x + 4 \sqrt { 2 }$
E) $y = 4 - \frac { \sqrt { 2 } } { 2 }$

Question

Quizplus · Accepted Answer

To convert the given polar equation $r \sin \left( \theta + \frac { \pi } { 4 } \right) = 4$ to rectangular form, we use the identities $r^2 = x^2 + y^2$, $r\cos(\theta) = x$, and $r\sin(\theta) = y$. The given equation can be rewritten using the sine addition formula: $\sin(\theta + \frac{\pi}{4}) = \sin(\theta)\cos(\frac{\pi}{4}) + \cos(\theta)\sin(\frac{\pi}{4})$. Since $\sin(\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, the equation becomes $r(\frac{\sqrt{2}}{2}y + \frac{\sqrt{2}}{2}x) = 4$, or $r\sqrt{2}(y + x) = 8$. Dividing both sides by $\sqrt{2}$ and rearranging, we get $y = -x + 4\sqrt{2}$, which matches choice B.

Convert to Rectangular Form rsin⁡(θ+π4)=4r \sin \left( \theta + \frac { \pi } { 4 } \right) = 4rsin(θ+4π​)=4

Convert to Rectangular Form $r \sin \left( \theta + \frac { \pi } { 4 } \right) = 4$