Find the maximum value of the expression $E=\sin\theta+\cos\theta+\sin2\theta$.
My approach is as follow ,let $E=\sin\theta+\cos\theta+\sin2\theta$, solving we get
$E^2=1+\sin^22\theta+\sin2\theta+2\sin2\theta(\sin\theta+\cos\theta)$ not able to approach from here.
Best Answer
Following @Batominovski's hint $$\left(\sin x+\cos x+\frac12\right)^2-\frac54=2\sin x\cos x+\sin x+\cos x$$ and the maximum value of $\sin x+\cos x$ is $\sqrt2$. Hence
$$\left(\sqrt2+\frac12\right)^2-\frac54=\sqrt2+1.$$
(The minimum is $-\dfrac54$ because the squared expression can vanish. There is also a local maximum with value $\left(-\sqrt2+\dfrac12\right)^2-\dfrac54=-\sqrt2+1$.)