What is the maximum value of $4(\sin x)^2 + 3(\cos x)^2$ The question is: What is the maximum value of: $4\sin^2\theta + 3\cos^2\theta$
This is the way I did it:
$4\sin^2\theta + 3\cos^2\theta = \sin^2\theta + 3\sin^2\theta + 3\cos^2\theta = \sin^2\theta + 3$
The max value of $\sin^2\theta$ is $1$, so the answer must be $4$. However my book says  that the answer is $5$. Where did I go wrong?

Here's a geometric "solution". With the curve $x(t)= \sqrt{3} \cos t$ and $y(t)=\sqrt{4} \sin t$, $0\leq t \leq 2 \pi$, we get an ellipse with major semiaxis of 2 along the y-axis and minor semiaxis of $\sqrt{3}$ along the x-axis. Then the distance squared to the origin $x(t)^2+y(t)^2=4\sin^2 t + 3 \cos^2 t$ must be maximized to the value of the major semiaxis squared. This is because we can imagine an ellipse inscribed inside a circle to see that it must be 4.
