Question

What are the components of the vector between the origin and the polar coordinate `(8, (5pi)/6)`?

Answer 1

`((-4sqrt3),(4))`

Explanation:

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

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

`color(orange)"Reminder"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(x=rcostheta , y=rsintheta)color(white)(a/a)|)))`

Here r = 8 and `theta=(5pi)/6`
`color(blue)"----------------------------------------"`
`rArrx=8cos((5pi)/6)`

`color(orange)"Reminder"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(cos((5pi)/6)=-cos(pi-(5pi)/6)=-cos(pi/6))color(white)(a/a)|)))`

`rArrx=-8cos(pi/6)=-8xxsqrt3/2=-4sqrt3`
`color(blue)"------------------------------------------------------------------"`

and y = `8sin((5pi)/6)`

`color(orange)"Reminder"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(sin((5pi)/6)=sin(pi-(5pi)/6)=sin(pi/6))color(white)(a/a)|)))`

`rArry=8sin(pi/6)=8xx1/2=4`
`color(blue)"----------------------------------------------"`

Thus the components of the vector are `((-4sqrt3),(4))`