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If $x = r$ ·sinθ·cosφ, $y = r$ ·sinθ·sinφ and $z = r$ ·cosθ, then $x^{2} + y^{2} + z^{2}$ equals:

Competition practiceTrigonometry

A $r^{2}$
B $2 r^{2}$
C $r^{2}$ si $n^{2}$ θ
D $r^{2}$ co $s^{2}$ θ

Step-by-Step Solution

x^{2} + y^{2} = r^{2}

si

n^{2}

θ(co

s^{2}

φ + si

n^{2}

φ) =

r^{2}

si

n^{2}

θ. Adding

z^{2} = r^{2}

co

s^{2}

θ gives

r^{2}

(si

n^{2}

θ + co

s^{2}

θ) =

r^{2}

. Both angles are eliminated.

Answer: A)

r^{2}

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