Question

How do you find the rectangular coordinates given the polar coordinates `(1/2, (3pi)/4)`?

Answer 1

`(-sqrt2/4,sqrt2/4)`

Explanation:

To convert from `color(blue)"polar to rectangular coordinates"`

That is `(r,theta)to(x,y)`

`color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(x=rcostheta , y= rsintheta)color(white)(2/2)|)))`

`"Here " r=1/2" and " theta=(3pi)/4`

`rArrx=1/2cos((3pi)/4)" and " y=1/2sin((3pi)/4)`

`color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(cos((3pi)/4)=-cos(pi/4)=-1/sqrt2)color(white)(2/2)|)))`

and `color(red)(bar(ul(|color(white)(2/2)color(black)(sin((3pi)/4)=sin(pi/4)=1/sqrt2)color(white)(2/2)|)))`

`rArrx=1/2cos(pi/4)=1/2xx-1/sqrt2=-1/(2sqrt2)=-sqrt2/4`

and `y=1/2sin(pi/4)=1/2xx1/sqrt2=sqrt2/4`

`rArr(1/2,(3pi)/4)to(-sqrt2/4,sqrt2/4)`