10. If $\tan A=\frac{\sqrt{2}}{4}$, where angle A terminates in quadrant three, determine the exact value of $\csc A$ and $\sec A$.

Step 1 ：We are given that \(\tan A = \frac{\sqrt{2}}{4}\) and that angle A terminates in quadrant three.

Step 2 ：The tangent of an angle is defined as the ratio of the sine to the cosine of that angle. Since we know the value of \(\tan A\), we can use this to find the values of \(\sin A\) and \(\cos A\).

Step 3 ：In the third quadrant, both sine and cosine are negative.

Step 4 ：The Pythagorean identity states that \(\sin^2 A + \cos^2 A = 1\). We can use this identity to find the values of \(\sin A\) and \(\cos A\).

Step 5 ：Once we have the values of \(\sin A\) and \(\cos A\), we can find the values of \(\csc A\) and \(\sec A\) by taking the reciprocals of \(\sin A\) and \(\cos A\) respectively.

Step 6 ：By calculation, we find that \(\sin A = -0.3535533905932738\) and \(\cos A = -0.9354143466934853\).

Step 7 ：Therefore, \(\csc A = -2.82842712474619\) and \(\sec A = -1.0690449676496976\).

Step 8 ：Final Answer: The exact values of \(\csc A\) and \(\sec A\) are \(\boxed{-2.82842712474619}\) and \(\boxed{-1.0690449676496976}\) respectively.