In order to transform the Polar coordinates of the point P(r,theta) into the Cartesian coordinates P(x,y), you must look at the right angled triangle made by the origin, P and P', where P' has coordinates (x,0) or (r,0).
In this triangle,
sin theta = y/r
and
cos theta = x/r,
therefore
x = r*cos theta
and
y=r*sin theta.
Your example has theta = (-21pi)/8 and r = -8, so
x = -8*sin((-21pi)/8)
y= -8*cos((-21pi)/8)
Using the property that the sin and cos functions are periodic with period 2npi, where n is an integer.
This means that for any a,
sin a = sin (a + 2npi)
cos a = cos(a+2npi).
sin((-21pi)/8) = sin((-16pi)/8 + (-5pi)/8) = sin(-2pi + (-5pi)/8)
So sin((-21pi)/8) = sin ((-5pi)/8).
sin((-5pi)/8) = sqrt(2+sqrt(2))/2
and
cos((-5pi)/8)=sqrt(2-sqrt(2))/2.
Back to the system of equations :
x=-4sqrt(2+sqrt(2))
y=-4sqrt(2-sqrt(2)).
