Answer: (a) \(A=11.3^\circ\) or \(108.4^\circ\). (b) \(B=-0.584\) or \(-0.00470\).

(a) Use

\(\operatorname{sec}^2 A=1+\tan^2 A.\)

Then

\(5(1+\tan^2A)+14\tan A-8=0.\)

Simplify:

\(5\tan^2A+14\tan A-3=0.\)

Factorise:

\((5\tan A-1)(\tan A+3)=0.\)

Thus

\(\tan A=\frac15\quad\text{or}\quad \tan A=-3.\)

For \(0^\circ\leq A\leq180^\circ\), this gives

\(A=11.3^\circ\quad\text{or}\quad A=108.4^\circ.\)

(b) From

\(5\sin\left(4B-\frac{\pi}{8}\right)+2=0,\)

we have

\(\sin\left(4B-\frac{\pi}{8}\right)=-\frac25.\)

Let

\(u=4B-\frac{\pi}{8}.\)

Since \(-\frac{\pi}{4}\leq B\leq\frac{\pi}{4}\),

\(-\frac{9\pi}{8}\leq u\leq\frac{7\pi}{8}.\)

The relevant solutions are

\(u=-0.4115\ldots\quad\text{or}\quad u=-2.7301\ldots.\)

Therefore

\(B=\frac{u+\frac{\pi}{8}}{4}.\)

This gives

\(\boxed{B=-0.00470\text{ or }B=-0.584}.\)