Question

If $x=r\sin A\cos C,y=r\sin A\sin C$ and $z=r\cos A$, then prove that $x^2+y^2+z^2=r^2$.

Answer 1

$x=r\sin A\cos C\Rightarrow x^2=r^2{\sin }^{2}A{\cos }^{2}C$

$y=r\sin A\sin C\Rightarrow y^2=r^2{\sin }^{2}A{\sin }^{2}C$

$z=r\cos A\Rightarrow r^2{\cos }^{2}A$

To Prove:- $x^2+y^2+z^2=r^2$

Proof:-

Taking L.H.S.-

$x^2+y^2+z^2$

$=r^2{\sin }^{2}A{\cos }^{2}C+r^2{\sin }^{2}A{\sin }^{2}C+r^2{\cos }^{2}A$

$=r^2{\sin }^{2}A({\cos }^{2}C+{\sin }^{2}C)+r^2{\cos }^{2}A$

$=r^2{\sin }^{2}A+r^2{\cos }^{2}A$

$=r^2({\sin }^{2}A+{\cos }^{2}A)$

$=r^2$

$=$ R.H.S.

Hence proved