Discussion
The minimum value of \(2\sin^2\theta + 3\cos^2\theta\) is:
Answer: B
We can rewrite the expression using the identity \(\sin^2\theta + \cos^2\theta = 1\).
\(2\sin^2\theta + 3\cos^2\theta = 2\sin^2\theta + 2\cos^2\theta + \cos^2\theta\)
= \(2(\sin^2\theta + \cos^2\theta) + \cos^2\theta\)
= \(2(1) + \cos^2\theta = 2 + \cos^2\theta\).
The value of \(\cos^2\theta\) ranges from 0 to 1. To get the minimum value of the expression, we take the minimum value of \(\cos^2\theta\), which is 0.
Minimum value = 2 + 0 = 2.
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