SOLUTION: If x = r sin A cos B and y = r sin A sin B and z = r cos A ,verify x^2 + y^2 + z^2 = r^2. Thanks so much in advance:)

Hi, there--

THE PROBLEM:
Given that 
x = r sin A cos B
y = r sin A sin B
z = r cos A

Verify x^2 + y^2 + z^2 = r^2.

A SOLUTION:
Substitute r sin A cos B for x, r sin A sin B for y, and r cos A for z in the equation.

x^2 + y^2 + z^2 = r^2
(r sin A cos B)^2 + (r sin A sin B)^2 + (r cos A)^2 = r^2

Simplify left-hand side.

r^2 (sin A)^2 (cos B)^2 +r^2 (sin A)^2 (sin B)^2 + r^2 (cos A)^2 = r^2

Factor r^2 from each term on the left-hand side.
r^2 [ (sin A)^2 (cos B)^2 + (sin A)^2 (sin B)^2 + (cos A)^2 ] = r^2

Factor (sin A)^2 from first two terms inside the brackets.

r^2 [(sin A)^2 [(cos B)^2 + (sin B)^2] + (cos A) ^2 ] = r^2

Simplify. (Recall that (cos B)^2 + (sin B)^2 = 1.)
r^2 [(sin A)^2 + (cos A) ^2 ] = r^2

Simplify again using same identity.
r^2 = r^2

VERIFIED!


Hope this helps! Feel free to email if you have any questions.

Mrs. Figgy
math.in.the.vortex@gmail.com