Question 826147
What we're going to do is take:
{{{x^2 + y^2 + z^2}}}
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^2}}}.<br>
Substituting in the given expressions:
{{{(r*sin(A)*cos(B))^2 + (r*sin(A)*sin(B))^2+(r*cos(A))^2}}}
Squaring each expression:
{{{r^2*sin^2(A)*cos^2(B) + r^2*sin^2(A)*sin^2(B)+r^2*cos^2(A)}}}
Factor out {{{r^2}}}:
{{{r^2(sin^2(A)*cos^2(B) + sin^2(A)*sin^2(B)+cos^2(A))}}}
Now, if we can find a way to turn the second factor into a 1, we will be finished! Factoring {{{sin^2(A)}}} out of the first two terms in the second factor:
{{{r^2(sin^2(A)(cos^2(B) + sin^2(B))+cos^2(A))}}}
Since {{{cos^2(B) + sin^2(B) = 1}}} this becomes:
{{{r^2(sin^2(A)+cos^2(A))}}}
Since {{{sin^2(A) + cos^2(A) = 1}}} this becomes:
{{{r^2}}}