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:)

Question 825898: 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:)
Answer by math-vortex(648) (Show Source): You can put this solution on YOUR website!

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

RELATED QUESTIONS
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. (answered by jsmallt9)

z=r(cos theta + i sin theta) show: a) {z}=r b) 1/z = 1/r(cos theta - i sin... (answered by stanbon)

If {{{x = (a) sin (theeta) - (b) cos (theeta)}}} and {{{y = (a) cos (theeta) + (b) sin... (answered by robertb,Alan3354)

Convert to polar: {{{ y = 4 }}} A.) {{{ r=4}}}/cosΘ B.) {{{ r=4}}}cosΘ C.)... (answered by MathLover1,Edwin McCravy)

Convert to polar: {{{y=4}}} A.) {{{r=4}}}/cosΘ B.) {{{r=4}}}cosΘ C.)... (answered by jim_thompson5910)

Graph each function and determine it's domain and range a) y = 2 sin ( x - π/4 ) +... (answered by Fombitz)

A curve with polar equation r = 10 6 sin θ + 25 cos θ r=106sinθ+25cosθ represents a... (answered by ikleyn)