The maximum value of $3 \cos \theta + 5 \sin \left( \theta - \frac{\pi}{6} \right)$ for any real value of $\theta$ is

If $\sin^2 \theta = \frac{x^2 + y^2 + 1}{2x}$,then $x$ must be

The maximum value of $(\cos \alpha_1) \cdot (\cos \alpha_2) \ldots (\cos \alpha_n)$ under the constraints $0 \leq \alpha_1, \alpha_2, \ldots, \alpha_n \leq \frac{\pi}{2}$ and $(\cot \alpha_1) \cdot (\cot \alpha_2) \ldots (\cot \alpha_n) = 1$ is

The maximum value of the function $f(x) = a \sin x + b \cos x$ is:

$\max_{0 \le x \le \pi} (16 \sin^2(\frac{x}{2}) \cos^3(\frac{x}{2}))$ is equal to:

Let $a$ be the maximum value of $(3 \cos \theta - 4 \sin \theta)$ and $\theta \neq \frac{n \pi}{2}$. If $\alpha = a \sin^2 \theta \cos^3 \theta$ and $\beta = a \sin^3 \theta \cos^2 \theta$,then $\sqrt{\frac{(\alpha^2 + \beta^2)^5}{(\alpha \beta)^4}} = $