If x = r sin A cos B, y = r sin A sin B and z = r cos A, then find x² + y² + z².
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A.
2r²
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B.
3/2 r²
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C.
r²
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D.
r²(cos² B + cos² A)
Correct Answer:
C. r²
Explanation:
To find the value of x² + y² + z², substitute the given expressions for x, y, and z:1. Calculate x² + y²:
x² + y² = (r sin A cos B)² + (r sin A sin B)²
x² + y² = r² sin² A cos² B + r² sin² A sin² B
Factor out r² sin² A:
x² + y² = r² sin² A (cos² B + sin² B)
Since cos² B + sin² B = 1, this simplifies to:
x² + y² = r² sin² A
2. Add z² to the result:
x² + y² + z² = r² sin² A + (r cos A)²
x² + y² + z² = r² sin² A + r² cos² A
Factor out r²:
x² + y² + z² = r² (sin² A + cos² A)
Since sin² A + cos² A = 1, the final result is:
x² + y² + z² = r²
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