Use a sketch to find the exact value of the following expression.
\[
\sec \left[\sin ^{-1}\left(-\frac{4}{7}\right)\right]
\]

Step 1 ：The problem is asking for the secant of the arcsine of -4/7. The arcsine function, also known as the inverse sine function, returns the angle whose sine is the given number. The secant function is the reciprocal of the cosine function.

Step 2 ：We need to find the angle whose sine is -4/7, then find the cosine of that angle, and finally take the reciprocal of that cosine value to get the secant.

Step 3 ：However, we can't directly calculate the cosine of the arcsine of a number. We need to use the Pythagorean identity, which states that \(\sin^2(x) + \cos^2(x) = 1\). We can rearrange this to find that \(\cos(x) = \sqrt{1 - \sin^2(x)}\).

Step 4 ：Since we know that \(\sin(x) = -4/7\), we can substitute this into the equation to find \(\cos(x)\). Then, we can find \(\sec(x)\) by taking the reciprocal of \(\cos(x)\).

Step 5 ：Substituting \(\sin(x) = -4/7\) into the equation \(\cos(x) = \sqrt{1 - \sin^2(x)}\), we get \(\cos(x) = \sqrt{1 - (-4/7)^2} = 0.8206518066482898\).

Step 6 ：Finally, we find \(\sec(x)\) by taking the reciprocal of \(\cos(x)\), which gives us \(\sec(x) = 1/\cos(x) = -1.2185435916898848\).

Step 7 ：Final Answer: The exact value of the expression \(\sec \left[\sin ^{-1}\left(-\frac{4}{7}\right)\right]\) is \(\boxed{-1.2185435916898848}\).