This point is in cylindrical form (r, theta, z)(r,θ,z). So let's first convert it to rectangular form (x,y,z)(x,y,z) by using the formulas x=rcos theta, y=r sin theta, z=zx=rcosθ,y=rsinθ,z=z
That is,
x=1*cos(pi/4)=1/sqrt2, y=1*sin(pi/4)=1/sqrt2, z=2x=1⋅cos(π4)=1√2,y=1⋅sin(π4)=1√2,z=2
hence, the point is (1/sqrt2, 1/sqrt2, 2)(1√2,1√2,2).
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 phiρ2=x2+y2+z2,x=ρsinϕcosθ,y=ρsinϕsinθ,z=ρcosϕ to change the point to spherical coordinates.
rho = sqrt ((1/sqrt2)^2 +(1/sqrt2)^2+(2)^2) = sqrt (1/2 + 1/2 + 4) = sqrt5 ρ=√(1√2)2+(1√2)2+(2)2=√12+12+4=√5
To find phiϕ let's use the formula z=rho cos phiz=ρcosϕ
Therefore,
z=rho cos phiz=ρcosϕ
2=sqrt 5 cos phi2=√5cosϕ
cos^-1(2/sqrt 5)=phicos−1(2√5)=ϕ
phi~~0.46ϕ≈0.46
:. (rho, theta, phi)~~(sqrt5, pi/4,0.46) ~~(2.24, 0.79,0.46)
