Range of the parameter of sphere intersecting with a plane

[Jadehaan]

1. The problem statement, all variables and given/known data
Find the range of the parameter d for which the intersection of the sphere x2+y2+z2=1 and the plane x+y+z=d is non-empty.


2. Relevant equations
Cartesian coordinates of a sphere:
x=rcos\thetasin\phi
y=rsin\thetasin\phi
z=rcos\phi


r=1

3. The attempt at a solution
I substitute x,y,z in both equations
d=sin\thetasin\phi+cos\phi+cos\thetasin\phi
cos2\thetasin2\phi+sin2\thetasin2\phi+cos2\phi=1
Since sin2\theta+cos2\theta=1
I get 1+cos2\phi=1
This implies that \phi=90
Which solves the first equation for d=sin\theta+cos\theta
Is this right?
Thanks for any help.

[LCKurtz]

I would think your answer would need to be in the form a ≤ d ≤ b. A simpler approach might be to observe the plane will intersect the sphere if the distance of the plane from the origin is ≤ 1. This can be easily done with vectors and no need for spherical coordinates.