Find the Least Value of 2 sin²θ + 3 cos²θ
Question:
\[ 2\sin^2\theta+3\cos^2\theta \]Find its least value.
Solution
Using the identity
\[ \sin^2\theta+\cos^2\theta=1 \]Write
\[ 2\sin^2\theta+3\cos^2\theta \] \[ =2(\sin^2\theta+\cos^2\theta)+\cos^2\theta \] \[ =2+\cos^2\theta \]Since
\[ 0\le \cos^2\theta \le 1 \]the minimum value of \(\cos^2\theta\) is \(0\).
Therefore,
\[ 2+\cos^2\theta \ge 2 \]Hence the least value is
\[ \boxed{2} \]