Minimizing trigonometric expression with the hint of substitution

Question

I have asked this question earlier for minimization of this expression.

$$ \left[\cos ^{2}\left(\dfrac{\theta }{2}\right) + \,\sqrt{\,{1 - \gamma}\,}\,\sin^{2}\left(\dfrac{\theta }{2}\right)\right] ^{2} + \dfrac{\gamma }{4}\,\sin^{2}\left(\theta\right)\quad \mbox{where}\quad 0 \le \gamma \le 1. $$

But then I am not sure if we can reach any relevant minimization (closed form) from the strategy mentioned here. What is the way to minize this with the following hint?

Hint is given in the first question itself.

Answer 1

Try starting with the half angle formulas for sine and cosine, so the entire expression will be with respect to $\theta$. Then multiply everything out and convert the $\sin^2 \theta$ to $1 - \cos^2 \theta$. After simplifying, you can differentiate with respect to $\theta$ and find the critical values.

Answer 2

You are minimizing$$\left(\frac{1+\cos\theta}2+\sqrt{1-\gamma}\,\frac{1-\cos\theta}2\right)^2+\frac\gamma4(1-\cos^2\theta).$$

After expansion, you get a quadratic trinomial in $\cos\theta$.

• find the location of the minimum by cancelling the derivative; if $\cos\theta$ falls in $[-1,1]$ it corresponds to the global minimum.

• otherwise, compute the trinomial for $\cos\theta=\pm1$ and keep the smallest value.