Question

if x = r cos alpha sin beta
y = r cos alpha cos beta
z = r sin alpha
show that x2 + y2 + z2 = r2

Answer 1

X^2+y^2+z^2=

R^2Cos^2alphasin^2beta + r^2cos^2alphacos^2beta+ R^2sin^2 alpha

R^2[Cos^2alpha(sin^2beta+ cos^2 beta)+sin^2 alpha]

R^2[Cos^2 alpha + sin^2 alpha]=r^2

Answer 2

Given that,

$x=r\cos\alpha\sin\beta$

$y=r\cos\alpha\cos\beta$

$z=r\sin\alpha$

We need to prove that,

$x^2+y^2+z^2=r^2$

Using L.H.S

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

Put the value into the formula

$(r\cos\alpha\sin\beta)^2+(r\cos\alpha\cos\beta)^2+(r\sin\alpha)^2$

$r^2(\cos^2\alpha(\sin^2\beta+\cos^2\beta)+r^2\sin^2\alpha$

Here, $\sin^2\beta+\cos^2\beta=1$

$r^2(\cos^2\alpha+\sin^2\alpha)$

$r^2$

R.H.S

Hence, This is required solution.