What's the difference between arccos(x) and sec(x)?

Question

I know this question was asked on this site, but I didn't understand the answer. Could someone give me the simplest explanation of this? (High school level explanation)

Answer 1

They are completely different.

$\sec x = \frac 1{\cos x} = 1\div \cos x$. This is the multiplicative reciprocal, which is sometimes call the multiplicative inverse.

$\arccos x$ is the function so that if $x = \cos y$ then $\arccos x$ is "going backwards" to get $y$ for which $x$ is the $\cos y$. so $\arccos x$ is defined as: If there are any $w$ so that $\cos w = x$ then one of those $w$ will be between $0$ and $\pi$; we define $\arccos x$ to be that $w$.

This is called the functional inverse.

That is it.

THE END.

...........................

Still reading? Well... this is probably why you got confused:

A multiplicative inverse of a value $K$ is a value $m$ so that $m \times K = 1$. In other words $m = \frac 1K$. We write the multiplicative inverse of $K$ as $\frac 1K$ but we also say $K^{-1}$ meaning $K$ raised to the negative $1$ power. (Remember $a^{-m} = \frac 1{a^m}$.) This will cause trouble later.

So the $\sec x = $ the multiplicative inverse of $\cos x$ means $\sec x = \frac 1{\cos x}$. Now we can write $\sec x = (\cos x)^{-1}$ !!!!!IF!!!!!! we mean "the value of $\cos x$ raised to the negative $1$ power.

The functional inverse of a function $f(x)$ is a function $g(x)$ so that $g(f(x)) = x$. And we often write the inverse function of $f(x)$ as $f^{-1}(x)$ but notice !!!!!! THIS IS VITALLY IMPORTANT AND YOU WILL !!!!DIE!!!! HORRIBLY !!!!!! IF YOU MISUNDERSTAND IT!!!!!! this does NOT mean $(f(x))^{-1}$ which is "the value of $f(x)$ raised to the negative $1$ power"; this, $f^{-1}(x)$, means "the function that 'reverses' $f$ and gets us back to where we started".

So $\arccos x$ is the functional inverse of $\cos x$. That is to say, $\arccos x$ is the function where $\arccos (\cos x) = x$. It "undoes" the $x \mapsto \cos x$ to get us back to $x$. We can write $\arccos x = \cos^{-1}(x)$ !!!!!!IF!!!!!! we mean "the function that 'reverses' $\cos$" and we !!!!!ABSOLUTELY UNDER RISK OF !!!!HORRIBLE!!!! DEATH!!!! !N!E!V!E!R! CONFUSE IT WITH !!!!!! $(\cos x)^{-1}$ meaning "raising $\cos x$ to the negative $1$ power.

It is VERY unfortunate that we use two very similar looking notation for two completely different concepts. It causes a lot of confusion in students.

(Actually logically the concepts are similar. $K^{-1} = \frac 1K$ undoes a multiplication whereas $f^{-1}(x)$ undoes a function. They both undo and get you back to where you started but one undoes multiplication. They other undoes a function.)

(I don't know. Maybe I shouldn't have mentioned that and maybe that's just more confusing. Forget I said anything.)

Answer 2

$\arccos(x)$ is the inverse function of $\cos(x)$ (restricted to the interval $[0,\pi]$)

$\sec(x)$ is the reciprocal, $\dfrac{1}{\cos(x)}$.

in otherwords, it is the difference between $f^{-1}(x)$ and $\dfrac{1}{f(x)}$.

Answer 3

$\arccos x$ is a value whose cosine is $x$, whereas $\sec x=\frac{1}{\cos x}$. The two kinds of "inverse" trigonometric functions can feel a bit esoteric when you're new to them, but let's pretend $\cos x=7x+12$. Then $\arccos x=\frac{x-12}{7}$ while $\sec x=\frac{1}{7x+12}$, which is clearly completely different.