# The least value of 2 sin

Question:

The least value of 2 sin2θ + 3cos2θ is ___________.

Solution:

$2 \sin ^{2} \theta+3 \cos ^{2} \theta$

$=2\left(\sin ^{2} \theta+\cos ^{2} \theta\right)+\cos ^{2} \theta$

$=2+\cos ^{2} \theta$

Since $-1 \leq \cos \theta \leq 1$

$\Rightarrow 0 \leq \cos ^{2} \theta \leq 1$

$\therefore 2 \sin ^{2} \theta+3 \cos ^{2} \theta \geq 2+0$

$2 \sin ^{2} \theta+3 \cos ^{2} \theta \geq 2$

i.e. least value of $2 \sin ^{2} \theta+3 \cos ^{2}$ is $2 .$