Convert the point given from Cartesian coordinates into Sphe

Convert the point given from Cartesian coordinates into Spherical coordinates and Cylindrical coordinates. Use the squareroot symbol \'squareroot\' where needed to give an exact value for your answer. Be sure to include parentheses where necessary, e.g. to distinguish 1/(2k) from 1/2k. Spherical: (3, 3 pi/4, 3 pi/4) rightarrow Cartesian: or Cylindrical:

Solution

We know that the cartesian coordinates corresponding to the spherical coordinates (,,Ø) are (x,y,z), where x = sinØcos, y = sinØ sin and z = cosØ. Here, = 3, = 3/4 and Ø = 3/4 . Then x = 2 sin(3/4) cos(3/4) =2*(1/2)*(-1/2) = -1 , y = 2*(1/2)*( 1/2)= 1 and z = 2 (-1/2) = -2. Thus, the cartesian coordinates corresponding to the given spherical coordinates are (-1, 1,-2).

We know that the cylindrical coordinates corresponding to the spherical coordinates (,,Ø) are (r,,z), where r = sinØ and z = cosØ. Here, = 3, = 3/4 and Ø = 3/4.Then r = 3sin(3/4)= 3*(1/2) = 3/2 and z = 3cos(3/4)= 3*(-1/2) = -3/2 . Further (as above) =3/4. Hence, the cylindrical coordinates corresponding to the given spherical coordinates are (3/2 , 3/4, -3/2)