MHB -11.7.94 Find the rectangular equation

[karush]

$\tiny{11.7.94 Kamahamai HS}$
Find the rectangular equation of the curve $r=\sin\left(\theta+\dfrac{\pi}{4}\right)$
$r=\sin \theta{\cos \dfrac{\pi}{4}
+{\cos \theta{\sin \dfrac{\pi}{4}}}}
=\sin \theta\left(\dfrac{\sqrt{2}}{2}\right)+\cos \theta\left(\dfrac{\sqrt{2}}{2}\right)
=\left(\dfrac{\sqrt{2}}{2}\right) (\sin \theta+\cos\theta)$

well so far anyway
Desmos plotted a circle

 

[skeeter]

$r = \dfrac{\sqrt{2}}{2}(\cos{t}+\sin{t})$

$r^2 = \dfrac{\sqrt{2}}{2}(r\cos{t}+r\sin{t})$

$x^2+y^2 = \dfrac{\sqrt{2}}{2}(x+y)$

which leads to …

$\left(x-\dfrac{\sqrt{2}}{4}\right)^2 + \left(y - \dfrac{\sqrt{2}}{4}\right)^2 = \left(\dfrac{1}{2}\right)^2$

 

[karush]

so that's how you get a circle
https://dl.orangedox.com/QS7cBvdKw55RQUbliE