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. Can you please help me?Thanks so much in advance:)

Algebra ->  Trigonometry-basics -> 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. Can you please help me?Thanks so much in advance:)      Log On


   

Question 826147: 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.
Can you please help me?Thanks so much in advance:)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
What we're going to do is take:
x%5E2+%2B+y%5E2+%2B+z%5E2
and substitute in the given expressions for x, y and z. Then we will use algebra and/or trig properties to transform the expression into r%5E2.

Substituting in the given expressions:

Squaring each expression:

Factor out r%5E2:

Now, if we can find a way to turn the second factor into a 1, we will be finished! Factoring sin%5E2%28A%29 out of the first two terms in the second factor:
r%5E2%28sin%5E2%28A%29%28cos%5E2%28B%29+%2B+sin%5E2%28B%29%29%2Bcos%5E2%28A%29%29
Since cos%5E2%28B%29+%2B+sin%5E2%28B%29+=+1 this becomes:
r%5E2%28sin%5E2%28A%29%2Bcos%5E2%28A%29%29
Since sin%5E2%28A%29+%2B+cos%5E2%28A%29+=+1 this becomes:
r%5E2