How do you find the rectangular coordinates of the point whose
spherical coordinates are #(4, 3π/4, π/3)#?

Question

Emma Aldridge · Accepted Answer

The answer is: #(sqrt2,sqrt6,-2sqrt2)#.

The formulas are:

#x=rsinthetacosphi#

#y=rsinthetasinphi#

#z=rcostheta#

So:

#x=4sin(3/4pi)cos(pi/3)=4*sqrt2/2*1/2=sqrt2#

#y=4sin(3/4pi)sin(pi/3)=4*sqrt2/2*sqrt3/2=sqrt6#

#z=4cos(3/4pi)=4*(-sqrt2/2)=-2sqrt2#.

Emma Aldridge · Answer

undefined To find the rectangular coordinates (x, y, z) of a point given its spherical coordinates (r, θ, φ), we use the following formulas:

$$ x = r \sin(φ) \cos(θ) $$
$$ y = r \sin(φ) \sin(θ) $$
$$ z = r \cos(φ) $$

Given the spherical coordinates (4, $ \frac{3π}{4} $, $ \frac{π}{3} $), we substitute these values into the formulas:

$$ x = 4 \sin\left(\frac{π}{3}\right) \cos\left(\frac{3π}{4}\right) $$
$$ y = 4 \sin\left(\frac{π}{3}\right) \sin\left(\frac{3π}{4}\right) $$
$$ z = 4 \cos\left(\frac{π}{3}\right) $$

After evaluating these expressions, we obtain the rectangular coordinates of the point.

How do you find the rectangular coordinates of the point whose spherical coordinates are #(4, 3π/4, π/3)#?