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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.
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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 the following trigonometric identities.
`sqrt((1 - cos theta)/(1 + cos theta)) = cosec theta - cot theta`
Prove the following trigonometric identities.
`(cot A + tan B)/(cot B + tan A) = cot A tan B`
If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ – 3 cos θ = ± 3.
Prove that `sqrt((1 + cos theta)/(1 - cos theta)) + sqrt((1 - cos theta)/(1 + cos theta)) = 2 cosec theta`
Prove the following identities:
`cosA/(1 - sinA) = sec A + tan A`
A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60° to 45° in 2 minutes. Find the speed of the boat in m/min.
Prove that `cot^2 "A" [(sec "A" - 1)/(1 + sin "A")] + sec^2 "A" [(sin"A" - 1)/(1 + sec"A")]` = 0
If `sec θ + tan θ = sqrt(3)`, complete the activity to find the value of sec θ – tan θ.
Activity:
`square = 1 + tan^2θ` ...[Fundamental trigonometric identity]
`square - tan^2θ = 1`
`(sec θ + tan θ) . (sec θ - tan θ) = square`
`sqrt(3) . (sec θ - tan θ) = 1`
`(sec θ - tan θ) = square`
Prove that `(sin^2θ)/(cos θ) + cos θ = sec θ`.
Simplify (1 + tan2θ)(1 – sinθ)(1 + sinθ)
