I was doing a bit of trigonometry, as I have been for a couple of years and it suddenly dawned on me that I don't really understand the trigonometric functions, at all.
You first learn the basic trig functions like this:
$$\sin(x) = \dfrac{o}{h}$$
$$\cos(x) = \dfrac{a}{h}$$
$$\tan(x) = \dfrac{o}{a}$$
Where $x$ is an angle and $o,a,h$ are the opposite, adjacent and hypotenuse sides respectively.
But I realized that I don't understand what the trig functions really , and especially why in god's name they work. This realization came when I had to sketch the functions $y=\sin^2(x)$ and I was dumbfounded; what exactly am I squaring? And why do the trig functions have a part below the x-axis? Why are they periodic?
Is there a way I can easily understand these functions intuitively?
Best Answer
As an example, the definition you're for $\sin$ using is:
$$\sin\theta=\frac{o}{h}$$
I think your confusion stems from the fact that the angle $\theta$ isn't mentioned in the right hand side of this equation, which means if you're just given the angle, it's unclear how to calculate the $\sin$ (because what's $o$ and $h$?). The idea is that you imagine a right triangle where one corner has an angle $\theta$, and then $\sin\theta$ is equal to $\frac{o}{h}$ applied to the sides of that triangle. "But wait, how do I know I'm imagining the 'correct' triangle?". That's the thing - it doesn't matter. As long as the angle $\theta$ is the same, the ratio $\frac{o}{h}$ will always be the same. That's what justifies defining $\sin$ in terms of a triangle, rather than directly in terms of an angle.
That said, the triangular definitions that you're given in middle school are terrible definitions for the $\sin$ and $\cos$ functions. Some of the problems are:
These definitions caused a lot of confusion for me up until late high school when I suddenly realized other definitions were available. The best definition for high school mathematics is as follows (it's almost the same as the one given by Shahab, just explained in a different way).
Let's say you've got an angle of $\theta$ jutting up from the horizon towards the sky. You extend a segment along this angle, of length $r$.
And now you want to know what the height and width of this segment are ($y$ and $x$ in the diagram). It turns out there's no easy formula for these, so mathematicians simply define new symbols $\sin\theta$ for the height, and $\cos\theta$ for the width. Actually, I was simplifying a tiny bit. You can imagine that the values of $y$ and $x$ don't just depend on the angle $\theta$. If you make $r$ longer, both $x$ and $y$ will get bigger, but $\theta$ won't change. So to account for that, $\sin$ is actually defined as the value $\frac{y}{r}$, and similarly for $\cos$. Hopefully you can see that these values only depend on $\theta$. If you keep $\theta$ the same but, say, double $r$, then $y$ will get doubled as well, so the ratio $\frac{y}{r}$ will still be the same.
If you think of the triangle formed by the segments $x$, $y$ and $r$ in the diagram, you can see that $r$ is the hypotenuse, $x$ is the adjacent side and $y$ is the opposite side, so you get back the old triangle definition. The difference is that now, you have a definition that also applies to angles greater than $90ç\circ$
You can now easily explain why the trig functions have a part below the $x$-axis. When the angle is greater than $180^\circ$ but smaller than $360^\circ$, where will the angle be pointing? It'll be pointing downwards, so the "height" of the angle will be negative, hence $\sin$ will be negative. Why are they periodic? Because when the angle hits $360^\circ$, it's like you're back to an angle $0$ again. See the animation in robjohn's answer for a great visualization.