If x-rsin-cos- -y-rsin-sin- - z-rcos- then the value of x2-y2-z2 is

The correct option is C r^2 Given: #160; x=rsinθcosϕ ,y=rsinθsinϕ , z=rcosθ Now, x^2 + y^2 + z^2 = r^2 sin^2 θcos^2ϕ + r^2 sin^2 θsin^2 ϕ ...

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Byju's Answer

Standard XII

Mathematics

Range of Trigonometric Expressions

If x=rsinθc...

Question

If $x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ$ then the value of $x^{2} + y^{2} + z^{2}$ is

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r^{2}

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Solution

The correct option is C $r^{2}$
Given: $x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ$
Now, $x^{2} + y^{2} + z^{2}$

= $r^{2} {sin}^{2} θ {cos}^{2} ϕ + r^{2} {sin}^{2} θ {sin}^{2} ϕ + r^{2} c o s^{2} θ$

= $r^{2} s i n^{2} θ ({cos}^{2} ϕ + {sin}^{2} ϕ) + r^{2} c o s^{2} θ$

= $r^{2} s i n^{2} θ + r^{2} c o s^{2} θ$

= $r^{2} (s i n^{2} θ + c o s^{2} θ)$

= $r^{2}$

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Similar questions

Q. If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then

(a)

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

(b)

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

(c)

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

(d)

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

If $x = r sin θ cos ϕ, y = r sin θ sin θ sin ϕ$ and $z = r cos θ$ , then $x^{2} + y^{2} + z^{2}$ is independent of
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Q. If

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If x=rsinθcosϕ,y=rsinθsinϕ,z=rcosθ then the value of x2+y2+z2 is

The correct option is C r2Given: x=rsinθcosϕ,y=rsinθsinϕ,z=rcosθNow, x2+y2+z2= r2sin2θcos2ϕ+r2sin2θsin2ϕ+r2cos2θ= r2sin2θ(cos2ϕ+sin2ϕ)+r2cos2θ= r2sin2θ+r2cos2θ= r2(sin2θ+cos2θ)= r2

If $x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ$ then the value of $x^{2} + y^{2} + z^{2}$ is