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Explanation:

Step1: Use double - angle formula

Recall $\cos(2t)=1 - 2\sin^{2}(t)$. Substitute it into the equation $8\cos(2t)=8\sin^{2}(t)+5$. We get $8(1 - 2\sin^{2}(t))=8\sin^{2}(t)+5$.

Step2: Expand the left - hand side

Expand $8(1 - 2\sin^{2}(t))$ to $8-16\sin^{2}(t)$. So the equation becomes $8 - 16\sin^{2}(t)=8\sin^{2}(t)+5$.

Step3: Rearrange the equation

Move all terms involving $\sin^{2}(t)$ to one side: $-16\sin^{2}(t)-8\sin^{2}(t)=5 - 8$. Combining like terms gives $-24\sin^{2}(t)=-3$.

Step4: Solve for $\sin^{2}(t)$

Divide both sides by $-24$: $\sin^{2}(t)=\frac{-3}{-24}=\frac{1}{8}$.

Step5: Solve for $\sin(t)$

Take the square root of both sides: $\sin(t)=\pm\sqrt{\frac{1}{8}}=\pm\frac{1}{2\sqrt{2}}=\pm\frac{\sqrt{2}}{4}$.

Step6: Find the values of $t$

When $\sin(t)=\frac{\sqrt{2}}{4}$, $t=\arcsin(\frac{\sqrt{2}}{4})\approx 0.36$ and $t=\pi-\arcsin(\frac{\sqrt{2}}{4})\approx 2.78$. When $\sin(t)=-\frac{\sqrt{2}}{4}$, $t=\pi+\arcsin(\frac{\sqrt{2}}{4})\approx 3.50$ and $t = 2\pi-\arcsin(\frac{\sqrt{2}}{4})\approx 5.92$.

Answer:

$0.36,2.78,3.50,5.92$