How can I conceptualize sin, cos, and tan?
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Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
Start with a unit circle, a circle of radius 1 centered at the origin (0, 0) of a plane. The rest is here:
https://www.mathsisfun.com/geometry/unit-circle.html
Say you have a circle with radius 1 at the center of the 2D plane, and you start at the rightmost point of it. If you go around this circle counterclockwise by say 55°, then sin 55° is the y-coordinate of your destination, cos 55° is the x-coordinate of your destination, and tan 55° is the slope of the line that goes from the origin to your destination. Same for other angles.
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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