Advertisements
Advertisements
प्रश्न
If x = r sin θ cos Φ, y = r sin θ sin Φ and z = r cos θ, prove that x2 + y2 + z2 = r2.
Advertisements
उत्तर
We have,
x = r sin θ cos Φ,
y = r sin θ sin Φ,
z = r cos θ
Squaring and adding,
x2 + y2 + z2
= r2 sin2θ cos2Φ + r2 sin2θ sin2Φ + r2 cos2θ
= r2 sin2θ (cos2Φ + sin2Φ) + r2 cos2θ
= r2 sin2θ x (1) + r2 cos2θ
= r2 (sin2θ + cos2θ)
= r2 x 1 = r2
Hence, x2 + y2 + z2 = r2.
Hence proved.
संबंधित प्रश्न
Prove the following trigonometric identities.
`cot theta - tan theta = (2 cos^2 theta - 1)/(sin theta cos theta)`
if `x/a cos theta + y/b sin theta = 1` and `x/a sin theta - y/b cos theta = 1` prove that `x^2/a^2 + y^2/b^2 = 2`
`(1 + cot^2 theta ) sin^2 theta =1`
`(cot^2 theta ( sec theta - 1))/((1+ sin theta))+ (sec^2 theta(sin theta-1))/((1+ sec theta))=0`
If cos A + cos2 A = 1, then sin2 A + sin4 A =
Prove the following identity :
`(secA - 1)/(secA + 1) = (1 - cosA)/(1 + cosA)`
Prove the following identity :
`(cotA + cosecA - 1)/(cotA - cosecA + 1) = (cosA + 1)/sinA`
Prove that sin θ sin( 90° - θ) - cos θ cos( 90° - θ) = 0
Prove that (cosec A - sin A)( sec A - cos A) sec2 A = tan A.
Prove the following identities.
sec6 θ = tan6 θ + 3 tan2 θ sec2 θ + 1
