Question

If \( x = 3 \sin \theta \), \( y = 3 \cos \theta \cos \phi \), \( z = 3 \cos \theta \sin \phi \), then \( x^2 + y^2 + z^2 = \)

• A. 18
• B. 27
• C. 9
• D. 3

Hint:

When dealing with trigonometric identities, simplify using known identities such as \( \cos^2 \theta + \sin^2 \theta = 1 \).

Answer 1

The Correct Option is C

Step 1: Use the given expressions for \( x \), \( y \), and \( z \).
We have: \[ x^2 = 9 \sin^2 \theta, \quad y^2 = 9 \cos^2 \theta \cos^2 \phi, \quad z^2 = 9 \cos^2 \theta \sin^2 \phi \] Adding these gives: \[ x^2 + y^2 + z^2 = 9 \sin^2 \theta + 9 \cos^2 \theta (\cos^2 \phi + \sin^2 \phi) \] Step 2: Simplify using trigonometric identity.
Using \( \cos^2 \phi + \sin^2 \phi = 1 \), we get: \[ x^2 + y^2 + z^2 = 9 (\sin^2 \theta + \cos^2 \theta) = 9 \] Step 3: Conclusion.
The correct answer is (C) 9.