Question

What are the components of the vector between the origin and the polar coordinate `(21, (-pi)/8)`?

Answer 1

`x = rho cos theta`
`y = rho sin theta`

Explanation:

`x = 21 cos (pi/8)`
`y=-21 sin(pi/8)`

`pi/8` is not an angle for which the value is universally known but you can find it with the trigonometric formula for the bisected angle:

`cos(pi/8) = cos (1/2*pi/4) = sqrt(((1+cos(pi/4))/2))= sqrt(((1+sqrt(2))/2)`
`sin(pi/8) = sin (1/2*pi/4) = sqrt(((1-cos(pi/4))/2))= sqrt(((1-sqrt(2))/2)`

and eventually:

`x = 21 sqrt(((1+sqrt(2))/2)`
`y=-21sqrt(((1-sqrt(2))/2)`