Question

What are the components of the vector between the origin and the polar coordinate `(-2, (5pi)/8)`?

Answer 1

`x=sqrt(2-sqrt2)=0.765367` and
`y=-sqrt(2+sqrt2)=-1.84776`

Explanation:

the x component

`x=r cos theta`
`x=-2*cos ((5pi)/8)`

but `cos ((5pi)/8)=-sin(pi/8)=-sqrt((1-cos(pi/4))/2)=-sqrt((1-1/sqrt2)/2)`

`cos ((5pi)/8)=-1/2sqrt(2-sqrt2)`

so that

`x=-2(-1/2sqrt(2-sqrt2))`

`x=sqrt(2-sqrt2)=0.765367`

the y component

`y=r sin theta`

`y=-2 sin ((5pi)/8)`

but `sin ((5pi)/8)=cos (pi/8)`

`cos (pi/8)=sqrt((1+cos(pi/4))/2)=sqrt((1+1/sqrt2)/2)=1/2sqrt(2+sqrt2)`

and

`y=-2*(1/2sqrt(2+sqrt2))`

`y=-sqrt(2+sqrt2)=-1.84776`

God bless America ....