In our prior work with inverse functions, we learned several important principles, including
The trigonometric function is periodic, so it fails the horizontal line test. Hence, considering this function on its full domain, it does not have an inverse function. At the same time, it is reasonable to think about changing perspective and viewing angles as outputs in certain restricted settings. For instance, we may want to say both
depending on the context in which we are considering the relationship between the angle and side length.It is also helpful to contextualize the importance of finding an angle in terms of a known value of a trigonometric function. Suppose we know the following information about a right triangle: one leg has length , and the hypotenuse has length . If we let be the angle adjacent to the side of length , it follows that . We naturally want to use the inverse of the cosine function to solve the most recent equation for . But the cosine function does not have an inverse function, so how can we address this situation?
While the original trigonometric function does not have an inverse function, we can instead consider a restricted version of the function that does. We thus investigate how we can think differently about the trigonometric functions so that we can discuss inverses in a meaningful way.
Consider the plot of the standard cosine function on with the portion on emphasized below.
For the cosine function restricted to the domain that we considered above, the function is strictly decreasing on its domain and thus passes the Horizontal Line Test. Therefore, this restricted version of the cosine function has an inverse function; we will call this inverse function the arccosine function.
In particular, we note that the output of the arccosine function is an angle. Recall that in the context of the unit circle, an angle measured in radians and the corresponding arc length along the unit circle are numerically equal. This is the origin of the “arc” in “arccosine”: given a value , the arccosine function produces the corresponding arc (measured counterclockwise from ) such that the cosine of that arc is .
For any function with an inverse function, the inverse function reverses the process of the original function. Thus, given , we can read this statement as saying “ is the cosine of the angle ”. Changing perspective and writing the equivalent statement, , we read this statement as “ is the angle whose cosine is ”. Just as and mean the same thing for a function and its inverse in general. To summarize, both expressions
mean the same thing for any angle that satisfies . We read as “ is the angle whose cosine is ” or “ is the inverse cosine of ”. Key properties of the arccosine function can be summarized as follows.
Just as the natural logarithm function allowed us to rewrite exponential equations in an equivalent way (for instance, and give the same information), the arccosine function allows us to do likewise for certain angles and cosine outputs. For instance, saying is the same as writing , which reads “ is the angle whose cosine is ”. Indeed, these relationships are reflected in the plot above, where we see that any point that lies on the graph of corresponds to the point that lies on the graph of .
Hence, is , and we now see that
Now, if you’re thinking, “Hey, we didn’t need that extra step!” Then you would be correct. But why didn’t we need that final step?
Let’s recall how we defined arccosine. Since cosine is a periodic function, it fails the horizontal line test. However, if we restrict cosine to a portion of its domain on which it is only decreasing, , then we may define a function on this domain such that for in . Arccosine is defined as the inverse of this function . Therefore, is the inverse of arccosine. Thus, in practice, cosine is the inverse of arccosine.
A word of caution: arccosine is only the inverse of restricted cosine, as we will demonstrate with the next example.
But wait!
Remember, we had to restrict the domain of cosine in order to define an inverse function, which we called arccosine. Arccosine is the inverse of the restricted cosine function, whose domain is . is larger than , so it is not within the domain of this restricted cosine.
Thus, we begin by simplifying .
Now, when we consider , we will once again recall the unit circle. We are looking at the upper hemisphere, but this time we want to find the angle in that corresponds to the point with -value .
Now, let’s look again at the graph of cosine. Here we highlight defined by , the restricted cosine function. We may use the symmetry of the graph of cosine to help find the appropriate values for arccosine.