Let $\theta$ be an angle in quadrant IV such that $\sin \theta=-\frac{7}{10}$. Find the exact values of $\sec \theta$ and $\tan \theta$.

Step 1 ：Let \(\theta\) be an angle in quadrant IV such that \(\sin \theta = -\frac{7}{10}\). This means that the opposite side of the right triangle is -7 and the hypotenuse is 10.

Step 2 ：We can use the Pythagorean theorem to find the adjacent side. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, \(adjacent = \sqrt{hypotenuse^2 - opposite^2} = \sqrt{10^2 - (-7)^2} = \sqrt{51} \approx 0.714142842854285\).

Step 3 ：Once we have the adjacent side, we can find \(\sec \theta\) and \(\tan \theta\). The secant of an angle in a right triangle is the ratio of the hypotenuse to the adjacent side, so \(\sec \theta = \frac{hypotenuse}{adjacent} = \frac{10}{\sqrt{51}} \approx 1.4002800840280099\).

Step 4 ：The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side, so \(\tan \theta = \frac{opposite}{adjacent} = \frac{-7}{\sqrt{51}} \approx -0.9801960588196068\).

Step 5 ：Final Answer: The exact values of \(\sec \theta\) and \(\tan \theta\) are \(\boxed{1.4002800840280099}\) and \(\boxed{-0.9801960588196068}\) respectively.