Q. 194.1( 9 Votes )

If x = r sin <spa A. x2 + y2 + z2 = r2

B. x2 + y2 z2 = r2

C. x2 y2 + z2 = r2

D. z2 + y2 x2 = r2

Answer :

Given: x = r sin θ cos ϕ, y = r sin θ sin φ and z = r cos θ, 

Solution:

x = r sin θ cos ϕ

Squaring both sides, we get

x2 = r2 sin2 θ cos2ϕ ……….(i)

and y = r sin θ sin ϕ

Squaring both sides, we get

y2 = r2 sin2 θ sin2ϕ ……….(ii)

z = r cos θ

Squaring both sides, we get

z2 = r2 cos2 θ ……….(iii)

Adding (i), (ii) and (iii), we get

x2 + y2 + z2 = r2 sin2 θ cos2ϕ + r2 sin2 θ sin2ϕ + r2 cos2 θ

= r2 (sin2 θ cos2ϕ + sin2 θ sin2ϕ + cos2 θ)

= r2 [sin2 θ (cos2ϕ + sin2ϕ) + cos2 θ]

∵ sin2 θ + cos2 θ = 1

= r2 [sin2 θ + cos2 θ] 

 Again apply the identity  sin2 θ + cos2 θ = 1

= r2 

Hence x2 + y2 + z2 = r2

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Trigonometric RatiosTrigonometric RatiosTrigonometric Ratios46 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses
RELATED QUESTIONS :

Prove the followiKC Sinha - Mathematics

Prove the followiKC Sinha - Mathematics

Prove the followiKC Sinha - Mathematics

If sin 77° = x, tKC Sinha - Mathematics

If cos55° = x<supKC Sinha - Mathematics

If cos40o</sKC Sinha - Mathematics