Find the value of $x$ satisfying the equation ${\sin }^{-1}(x-1)={\tan }^{-1}\left(3\right)$.

Given that $\theta$ is an acute angle measured in radians and that $\cos \theta =k$. Find, in terms of $k$, an expression for:

$\sin \theta$

Given that $\theta$ is an acute angle measured in radians and that $\cos \theta =k$. Find, in terms of $k$, an expression for: $\tan \theta$.

Given that $\theta$ is an acute angle measured in radians and that $\cos \theta =k$. Find, in terms of $k$, an expression for: $\cos (\pi -\theta )$

Solve the equation ${\cos }^{-1}\left(8x^4+14x^2-16\right)=\pi$

Solve the equation $\sin 2x=5\cos 2x$ for $0°\le x\le 180°$.

Solve the equation $\frac{13{\sin }^2\theta }{2+\cos \theta }+\cos \theta =2$ for $0°\le x\le 180°$.

Solve the equation $2{\cos }^{2}x=5\sin x-1$ for $0°\le x\le 180°$.

Sketch, on a single diagram, the graphs of $y=\cos 2\theta$ and $y=\frac{1}{2}$ for $0\le x\le 2\mathrm{\pi }$.