For the following trigonometric equation, find the radian measure of $A$ in terms of $\pi$.

$\sin \left(2A\right)=\cos \left(2A\right),0\le A<2\pi$

Solve the equation ${\cos }^{2}3x=\frac{1}{4}$ within the interval $-\pi \le x\le \pi$.

A flagpole at a schoolyard extends cables to points $G$ and $H$ on the flat ground, where $H$ is $7\text{ m}$ closer to the base of the flagpole than $G$. The cable to point $G$ forms an angle of ${40}^{\circ }$ with the horizontal and the cable to point $H$ forms an angle of ${65}^{\circ }$ with the horizontal. What is the height of the flagpole? Round your answer to two decimal places.

A right triangle $PQR$ has a right angle at $R$. The sides $p$, $q$, and $r$ are opposite the angles $P$, $Q$, and $R$, respectively. Calculate the lengths of sides $p$ and $q$ if $r=5$ and $Q=\frac{\pi }{6}$.

Rewrite the expression in terms of exponentials and simplify: $\sinh \left(4x\right)+\cosh \left(4x\right)$