Maximum of $\sqrt{\frac{1}{4}\cdot \sin^2(t)+\sin^2(t+\frac{\pi}{3})}$

Question

Maximum of $\sqrt{\frac{1}{4}\cdot\sin^2(t)+\sin^2(t+\frac{\pi}{3})}$

In my opinion, the maximum of the sinus is 1, so I calculated $\sqrt{\frac{1}{4}\cdot1+1}$

This is wrong, why?

Answer 1

You are right about the sine. However, the max for $\sin t$ and the max for $\sin(t+\pi/3)$ do not occur at the same value of $t$, so these expressions can't be maximized separately.

Answer 2

Note that the two values for $\sin^2$ don't occur for the same value of $t$, indeed

$$\sin^2(t)=1\implies t=\frac{\pi}2+k\pi \implies \sin^2(t+\frac{\pi}{3})=\frac14$$

To evaluate the maximum let observe that

$$\left(\sqrt {f(x)}\right)'=\frac{f'(x)}{2\sqrt {f(x)}}$$

then it suffices to evaluate

$$f(t)=\frac{1}{4}\sin^2(t)+\sin^2(t+\frac{\pi}{3})\implies f'(t)=\frac12\sin t \cos t+2\sin(t+\frac{\pi}{3})\cos(t+\frac{\pi}{3})=0$$

Answer 3

You've definitely calculated an upper bound for the function. The issue is that $ \sin(t) $ and $ \sin(t + \frac\pi 3 ) $ cannot both equal $ 1 $ at the same time (i.e. for the same value of $ t $).

To find the exact maximum, you'll need to differentiate the function (using the chain rule a couple of times).

Answer 4

Other people have pointed out the OP's error. Now, I am supplying a solution.

Note that $$\sin\left(t+\frac{\pi}{3}\right)=\frac{1}{2}\sin(t)+\frac{\sqrt{3}}{2}\cos(t)\,.$$ That is, $$\begin{align}\frac{1}{4}\sin^2(t)+\sin^2\left(t+\frac{\pi}{3}\right)&=\frac{\sin^2(t)+\big(\sin(t)+\sqrt{3}\cos(t)\big)^2}{4}\\&=\frac{5}{8}+\frac{2\sqrt{3}\sin(2t)+\cos(2t)}{8}\,.\end{align}$$ We then apply the Cauchy-Schwarz Inequality to get $$-\frac{\sqrt{13}}{8}\leq \frac{2\sqrt{3}\sin(2t)+\cos(2t)}{8}\leq +\frac{\sqrt{13}}{8}\,.$$ Consequently, $$\sqrt{\frac{5-\sqrt{13}}{8}}\leq \sqrt{\frac{1}{4}\sin^2(t)+\sin^2\left(t+\frac{\pi}{3}\right)}\leq \sqrt{\frac{5+\sqrt{13}}{8}}\,.$$ It is not difficult to see that both the inequality on the left-hand side and the inequality on the right-hand side are sharp.