We all know that sine is one over cosecant, cosine is one over secant, etc.
But is there any relationship between sine and cosine, secant and cosecant, and tangent and cotangent?
What I am asking here is the meaning of the "co-" prefix.
Thanks.
[Math] relationship between trigonometric functions and their “co” functions
terminologytrigonometry
Related Solutions
Congratulations! You've stumbled in to a very interesting question!
In higher mathematics, we often notice that some things which are really easy to talk about but difficult to express rigorously have a property which is really easy to express rigorously but something that we probably wouldn't have thought of to begin with.
The trig functions are one of these things. With (a lot of) effort, you can show that
$$\sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \frac{x^9}{362880} - \cdots $$
where the patterns of increasing the powers of $x$ by $2$, and switching between $+$ and $-$ signs continues forever. (The denominators also have a pattern: take the power that $x$ is raised to in the term and multiply it by all of the smaller numbers down to $1$; that is the number in the denominator). Note that you have to use radians for this exact formula to work; of course you could come up with one for degrees as well.
When you start realizing that circles are actually quite tricky objects to define, formulas like that one start to look more appealing. I have had multiple mathematics textbooks take this infinitely long expression as the definition of the sine function. (It turns out to be the same thing as the circle definition, but… well, circles get complicated.)
Of course, we can't sit around multiply and add for the rest of our lives just to compute sin $1$, but we can just cut off the operations after a couple terms. If you go out to the $x^7$ term, you can guarantee that your answer is accurate to at least 3 decimal places as long as you use angles between $-\frac{\pi}{2}$ and $\frac\pi 2$. (These are the only angles you really need, if you get rid of multiples of $\pi$ properly.)
The cosine formula, in case you are interested, is similar: $$\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720}+ \frac{x^8}{40320}-\cdots$$
The internet has formulas for the other trig functions, but you can always just combine these.
As copper.hat says, there are also these large books where people did the calculations once and wrote them down so that nobody would have to do them again. Of course, these were made long before computers existed; nobody makes them anymore! But somebody from your parents' or grandparents' generation probably still has one sitting in their house.
It depends on how you look at it I guess, but:
$$\cot(x) = \frac{1}{\tan(x)}$$
$$\csc(x) = \frac{1}{\sin(x)}$$
$$\sec(x) = \frac{1}{\cos(x)}$$
So the three "extra" functions your friend told you about are just derived from the three you know. But if that's the rule, then two of the ones you know,
$$\cos(x) = \sin\left(\frac{\pi}{2} - x\right)$$
$$\tan(x) = \frac{\sin(x)}{\cos(x)} = \frac{\sin{x}}{\sin\left(\frac{\pi}{2} - x\right)}$$
are also just derived functions. Hence we would say there is only one trigonometric function, for example $\sin{x}$.
(As others have mentioned, this statement works even counting hyperbolic functions, because of properties like $\cosh(x) = \cos(ix)$ and so on, or using $e^{i\theta} = \cos{\theta} + i\sin{\theta}$. But since you don't appear to be at this level of math yet, I won't go into detail about that.)
Bottom line: We only need one trigonometric function, but for practical reasons, there are more.
Best Answer
There is; they are the function applied to complement of the given angle:
$$ \cos(x) = \sin(\frac{\pi}{2} - x) $$
$$ \csc(x) = \sec(\frac{\pi}{2} - x) $$
$$ \cot(x) = \tan(\frac{\pi}{2} - x) $$
These hold for all $x$, not just for $x < \frac{\pi}{2}$. Also note that these relationships hold in the reverse direction, since taking the complement of a complement results in the original angle.