Why is the Arccosine of $30$ degrees undefined?

Question

Recently, while working on Trigonometry, a problem came up in which I was asked to evaluate the value of $\cos(\arccos(30^\circ))$, and I stated that the value of this function was $30$ degrees (because the cos and arccos cancel each other out). The answer, however, was stated to be undefined, as the value of $\arccos(30)$ was undefined. I plug this into Desmos, and Desmos seems to agree with that assertion, but then I plug it into Symbolab, and Symbolab seemed to agree with my answer. What is the reason for this

Answer 1

The correct answer would be undefined, as you pointed out. The key to solving this problem is understanding the input and output of trigonometric and inverse trigonometric functions.

Your assumption that the $\cos()$ and $\arccos()$ functions would cancel out is incorrect for this particular case, but when used in a different order or with different input values, might hold. I'll explain this later on in my answer.

Consider the $\cos()$ function: it takes an angle as an input (in degrees or radians) and then outputs a value that ranges between $1 \leq \cos(x) \leq 1$.

Consider the $\arccos()$ function: it takes a value between $[-1,+1]$ and outputs an angle in degrees or radians (based on how you set up a solver to produce the output).

Now, let's take a look at your question: $\cos(\arccos(30^\circ))$. Break this problem down, first. You need to find the inverse cosine of $30^\circ$ meaning that if I was to feed this unknown value into a regular $\cos()$ function, it would yield $30^\circ$. But this goes against the very definition of a $\cos()$ function that I've laid above. That's the whole reason $\arccos()$ cannot be defined for $30^\circ$, because it's meant to take this value as an input and can only process inputs between $[-1,+1]$. Hence, the answer is undefined.

Now, when would your assumption that the trigonometric and inverse trigonometric functions cancel? This is subject to the conditions imposed on these functions, specifically the range in which their outputs and inputs are defined.

Consider, $\arccos(\cos(30^\circ))$. In this case, solving the internal parentheses yields a value of $\frac{\sqrt3}{2}$. If you plug this as input into the $\arccos()$ function, it yields $30^\circ$ as the answer. In this case, note how the inputs and outputs match the acceptable values that these functions can take/put out.

As for the reason why different solver engines yield different outputs, it is primarily dependent on how they are built to handle undefined answers. In some cases, they often correct user errors by switching the order of functions in a chain equation, and it is apparent that $Symbolab$ uses this implementation. However, it appears that $Desmos$ accurately represents the given equation as undefined.

Answer 2

Of course, cosine and arccosine, as functions, have nothing to do with angles whether in degrees or radians. Like any functions, they take a number as input and output a number. Of course, while the domain of cosine is all real numbers, the range is from -1 to 1 so the domain of arccosine is -1 to 1 and so arccos(30) is undefined.

But that is over the real numbers. If we extend cosine to the complex numbers, then we have $cos(x)= \frac{e^{ix}+ e^{-ix}}{2}$ and it may have values outside of -1 to 1. For $cos(x)= 30$ we need to solve $\frac{e^{ix}+ e^{-ix}}{2}= 30$.

Then $e^{ix}+ e^{-ix}= 60$. Let $y= e^{ix}$ so we have $y+\frac{1}{y}= 60$. Multiply both sides by y to get $y^2+ 1= 60y$ or $y^2- 60y+ 1= 0$. By the quadratic formula, $y= e^{ix}= \frac{60\pm \sqrt{3600- 4}}{2}= 30\pm\sqrt{899}$. Taking the positive square root, $e^{ix}= 30+ \sqrt{899}$ is approximately 60 so ix is approximately ln(60)= 4.0943445622221006848304688130651 and x= arccos(30) is approximately -4.0943445622221006848304688130651i.