How do you convert (1, pi/4, 2) into spherical form?

1 Answer
Sep 12, 2017

see below

Explanation:

This point is in cylindrical form (r, theta, z). So let's first convert it to rectangular form (x,y,z) by using the formulas x=rcos theta, y=r sin theta, z=z

That is,

x=1*cos(pi/4)=1/sqrt2, y=1*sin(pi/4)=1/sqrt2, z=2

hence, the point is (1/sqrt2, 1/sqrt2, 2).

Now let's use the formulas
rho^2 = x^2 + y^2 +z^2, x=rho sin phi cos theta, y=rho sin phi sin theta, z=rho cos phi to change the point to spherical coordinates.

rho = sqrt ((1/sqrt2)^2 +(1/sqrt2)^2+(2)^2) = sqrt (1/2 + 1/2 + 4) = sqrt5

To find phi let's use the formula z=rho cos phi

Therefore,

z=rho cos phi

2=sqrt 5 cos phi

cos^-1(2/sqrt 5)=phi

phi~~0.46

:. (rho, theta, phi)~~(sqrt5, pi/4,0.46) ~~(2.24, 0.79,0.46)