Advertisements
Advertisements
Question
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
Solution
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.
RELATED QUESTIONS
Prove that ` \frac{\sin \theta -\cos \theta +1}{\sin\theta +\cos \theta -1}=\frac{1}{\sec \theta -\tan \theta }` using the identity sec2 θ = 1 + tan2 θ.
Prove the following trigonometric identities.
(sec A − cosec A) (1 + tan A + cot A) = tan A sec A − cot A cosec A
Prove that:
`(tanA + 1/cosA)^2 + (tanA - 1/cosA)^2 = 2((1 + sin^2A)/(1 - sin^2A))`
Prove that:
`(sinA - cosA)(1 + tanA + cotA) = secA/(cosec^2A) - (cosecA)/(sec^2A)`
`sqrt((1+cos theta)/(1-cos theta)) + sqrt((1-cos theta )/(1+ cos theta )) = 2 cosec theta`
`(sin theta +cos theta )/(sin theta - cos theta)+(sin theta- cos theta)/(sin theta + cos theta) = 2/((sin^2 theta - cos ^2 theta)) = 2/((2 sin^2 theta -1))`
If `sqrt(3) sin theta = cos theta and theta ` is an acute angle, find the value of θ .
Prove that:
`(sin^2θ)/(cosθ) + cosθ = secθ`
Prove the following identity :
`(1 + cotA)^2 + (1 - cotA)^2 = 2cosec^2A`
Prove the following identity :
`tan^2θ/(tan^2θ - 1) + (cosec^2θ)/(sec^2θ - cosec^2θ) = 1/(sin^2θ - cos^2θ)`
