Trigonometry confusion

Question

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?

Answer 1

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:

1. They only work for angles $< 90^\circ$. There can't be any oblique angles in a right triangle, so you can't talk about a hypotenuse.
2. When working with polar coordinates, you have to perform complicated visualizations, superposing triangles over everything in your head in order to understand where the formulae are coming from.
3. They obfuscate the relationship between $\sin\theta$ and the angle $\theta$.
4. It's less clear why they're worth studying. Who cares about ratios of sides in right angle triangles?

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.

Answer 2

There are several ways to give meaning to $\sin$. Here are a couple: $$\sin(x):=\sum \limits_{n=0}^{+\infty}\left(\dfrac{(-1)^n}{(2n+1)!}x^{2n+1}\right),\text{ for all }x\in \Bbb R.$$

You can also define it as the periodic extension of the natural extension of the inverse of $\arcsin\colon [-1,1]\to [-\pi /2,\pi/2]$ to $[-\pi, \pi]$, this of course begs for a definition of $\arcsin$. You can define it as $\displaystyle x\mapsto \int \limits _{-1}^x\dfrac{1}{\sqrt{1-x^2}}\mathrm dx-\dfrac \pi 2$, for all $x\in [-1,1]$.

Answer 3

Here is one way to define the $\sin$ function (and by extension all the other trigonometric functions). Take a circle in the cartesian plane whose radius is 1 and center is $(0,0)$. Starting with the point $(1,0)$ imagine there is an ant placed on it, which is constrained to move only on the circle. For $x\ge 0$ we define $\sin x$ as the y-coordinate of the point which the ant has reached after travelling $x$ distance in the counter-clockwise direction. For $x<0$ it is y-coordinate which the ant reaches on travelling in the clockwise direction.

It would be an interesting exercise for you to see how this definition tallies with the one you have read. (Hint: Think similar triangles). How this definition is equivalent to the infinite series definition in another answer has to deal with the existence and uniqueness theorem of differential equations.

With this definition it is also clear why in $[0,2\pi)$ the values of $\sin$ are negative after the ant has traveled $\pi$ distance (for then the y-coordinates start becoming negative) and why $\sin$ repeats after $2\pi$ (for then the ant is simply retracing its path).

A similar logic with the hyperbola instead of the circle leads to hyperbolic functions.

Answer 4

Imagine a small object that is fixed at the edge of a turntable, and the turntable is made to turn counterclockwise at a constant rate of one radian per unit time. Imagine that looking down at the turntable, a coordinate system is set up so that the center of the turntable is placed at the origin, and that the object starts at the point with coordinates $(1, 0)$. Then the sine function measures the $y$-coordinate of the object as a function of time as the turntable rotates, and the cosine function measures the $x$-coordinate. You can see that at various times the object will be "below" the center or origin (will have $y$-coordinate less than $0$); this is where the sine $\sin(t)$ is negative. Similarly, at times when the object is to the left of the center, the cosine $\cos(t)$ is negative.

It's not a bad idea to practice visualizing this in various ways until the graph of the sine function becomes intuitively clear. For example, imagine that a ball is affixed to the edge of a very thin plate, and from behind a camera the plate is held up and observed edgewise, so that it looks like a vertical line segment. Again the plate is made to turn at a constant rate, and a light from directly behind the camera is turned on so that the ball casts a shadow on the wall (and the shadow of the plate is a thin vertical line as before). Then the shadow of the ball looks like it's bobbing straight up and down. Now imagine the whole ensemble (light, camera, plate, and ball) moving to the right at constant velocity, while a second camera records the trajectory of the ball's shadow on the wall. That trajectory will assume the shape of a sine curve.

Trigonometric functions are often called circular functions because they measure uniform circular motion in just such a way.

Answer 5

$(\cos(\theta),\sin(\theta))$ is the position of the point on the unit circle whose angle (measured counterclockwise from right) is $\theta$.

$\hspace{3.5cm}$

Note how each changes sign (and color) and repeats every $2\pi$.