Solution

To simplify these nested functions, we need to take things one step at a time. Begin by simplifying the inner function. In the case of the first problem:

\[ \cos\left(\frac{7\pi}{{6}}\right)=-\frac{\sqrt{ 3 } }{ 2 } \]

Once we have the inner function simplified, we look at the original problem with the simplified inner portion:

\[\cos^{-1}\left(-\frac{\sqrt{ 3 } }{ 2 }\right) \]

This now takes us back to the questions from the previous examples, where we needed to reference the inverse trig semi-circles. Notably, for the Cosine function, the angle which corresponds to \(-\frac{\sqrt{ 3 } }{ 2 }\) is \(\frac{5\pi}{{6}}\). Thus, our answer is as follows:

\[ \cos^{-1}\left(\cos\left(\frac{7\pi}{{6}}\right)\right)=\frac{5\pi}{{6}} \]

This answer is confusing to many students who are new to the Inverse Trig functions because it seems as if the function and its inverse should "cancel out" and the answer ought to be the \(\frac{7\pi}{{6}}\) we see in the original question statement. The key to understanding why it is not is to remember that Cosine/Cosine Inverse function only operate on a very specific domain/range, namely on the \([0,\pi]\) and \([-1,1]\) domain/range of Cosine. This means that \(\frac{7\pi}{{6}}\) simply isn't in the range of the inverse Cosine! While strange, the lesson I suggest you take away from this is to be very careful of simplifying inverse trig functions because you must be very aware of the Domain/Range of each inverse trig.

Moving on to the second question: what does \(\cos^{-1}\left(\frac{ 4 }{ 5 }\right)\) mean? I find it most helpful to write it out like this:\[\theta = \cos^{-1}\left(\frac{ 4 }{ 5 }\right)\]I do this to strongly remind myself that the output of any inverse trig function is an angle. The \(\frac{{4}}{{5}}\), then, is the \(x\) coordinate of a point on the Unit Circle, so we can draw the right triangle:

Cosine Circle Here

From this, we can use the Pythagorean Theorem to solve for the missing side as necessary: \(\left(\frac{{4}}{{5}}\right)^2+y^2=1\) will give us \(y=\frac{{3}}{{5}}\). In this example, we don't necessarily need to do so, but you should always keep in mind that it is possible.

What is important, in this image, is that the point on the Unit Circle is already on the modified semi-circle that is required for the inverse trig to be understood. Namely, we can see that, whatever angle \(\cos^{-1}\left(\frac{{4}}{{5}}\right)\) is, it must be in the first quadrant. Therefore, the Cosine of that angle must be exactly the \(x\) coordinate of the point on the unit circle, \(\frac{{4}}{{5}}\). Thus, in this example, the Cosine and Cosine Inverse do appear to "cancel out" leaving us with just the inner value:

\[ \cos\left(\cos^{-1}\left(\frac{{4}}{{5}}\right)\right)=\frac{ 4 }{ 5 } \]