How can I conceptualize sin, cos, and tan?
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Take a right-angle triangle.
You’ve got three lines of different lengths, and they join at three different angles – so there’s six different facts defining the triangle in total.
The relationships between the lengths and the angles are well-defined, meaning you can use a subset of those six facts to fill in missing ones.
One relationship you probably know already is Pythagoras’ theorem: A^2 + B^2 = C^2. If you know the length of two sides, you can work out the length of the third.
But as it turns out, there’s three more-complex functions that map between the angle at one of the corners, and *the ratio of the lengths of two of the sides*. What this means is, once you have one angle and one length, you can get a length-ratio. You can then multiply your known side by that ratio to get a second length, and then use Pythagoras to get the third, and so know everything about the triangle.
Actually calculating those functions, turning an angle into a ratio, is a whole page of arithmetic – but it’s just just drudge work, adding up an infinite series of terms until you get the precision you want. Luckily, it’s trivial for computers/calculators, or you can look up the answer in a table.
Now, the fun part is that these functions have broader implications than just calculating the height of a flag pole.
If you have a triangle with a hypoteneuse of 1, then the other two sides are going to be sin(θ) and cos(θ).
[Draw that triangle inside a unit circle](https://i.gifer.com/8O8I.gif), the hypoteneuse will be the radius, and sin/cos will be the height/width of the segment it describes.
Think of the X position of a horse on a merry-go-round – it doesn’t just sweep linearly from side to side; when it’s at the sides, its movement is all forward/back so it barely changes horizontally, but when it’s in the middle, its movement is all horizontal, so it changes more per second. Plot that position on a graph, you get a sine wave.
Any time you have to with deal angle vs position – anything involving rotation or oscillation – sin/cos/tan are involved.
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