Teach me Trigonometry

[monkey0506]

My only experience with trig was in Pre-Calculus in high school, 10 years ago. I am now finally enrolled in college, and I am considering taking trigonometry over the course of a 3 week mini-mester, starting next month. In preparation for that, teach me everything about trig, sage AGSers!

Thanks!

[Snarky]

Congrats on matriculating!

Although trigonometry sounds like it's all about triangles, it's actually mostly about circles, namely the relationship between the radius and angle on the one hand, and the x-coordinate and the y-coordinate on the other, as you trace a circle of a certain size around the origin.

[Gurok]

For games programming, the only functions I ever seem to need are Arctan (capital A) and the Pythagorean theorem.

sec(x) is 1/cos(x), but cosec(x) is 1/sin(x)
Would've been easier to remember if it was cos paired with cosec instead

Finally, here's a study aid: http://sineridergame.com/SineRider.html

Edit: Another tricky thing is quadrants

If it's in the second quadrant, you need to do pi minus the angle to get it into the first quadrant so you can work with it
If it's in the third, it's the angle minus pi
If it's in the fourth, it's 2pi minus the angle

[monkey0506]

Well it seems my years of honing my algebra skills in programming random things has been ineffably useful. I only have two days left of the class, and then the final exam, but I haven't made a single grade lower than 96 I think on any assignments, with the exception of one PowerPoint presentation that I didn't spend enough time on, which I made a 90 on.

In short, I'll probably only need to make a 30 or 40 (percent) on the final exam to get an A for the course. It makes a hell of a lot of sense to me, which my roommate nearly killed me over (they having dropped the class in the Spring semester).

Cheers!

[ollj]

I prefer geometry over trigonometry, mostly because you just have to confuse the arcustangent with the arccosine once to be screwed up forever.
If i can solve something trigonometric with vectors, i prefer thinking in vectors!
It is just so much easier to visualize any problem with 3 pencils and a few right-angles than to deal with pi and complex numbers, at least intuitively to solve any abstract problem.

as far as geometry goes, whenever i need to remember a formula to calculate any intersection/distance/angle i just go here, the script is not in C, but the functions and math are pretty universal.


--a dregree in balooney--

when it comes to calculating angles, you just have to start thinking in radians as soon as possible and not even bother EVER to think in degrees.
degrees are just a confusing outdated notation, just think of them as 2 currencies, "degree" is like an ancient roman coin, and "radians" are like cryptocurrencies.

they are exactly the same as math are concerned, just with a different factor, and the exchange value is
0.017453292519943295769236907684886f RADIANT == 1 DEGREE
//which equals //
1 RADIANT == 57.295779513082320876798154814105f DEGREE

that conversion-rate between 2 scales, of whom the older dumber one (DEGREES) just used to work better with integers in base 60 (as it was used by ancient astronomers back then), matters just like knowing the speed of light and other universal constants matters.
using "degree" is like using inches to measure distance, just don't to it on the fundamental level of maths, especially when calculating angles in trigonometry!
you must only think in radians, because radians are so much easier in relation to Pi and imaginary numbers, you don't have to convert the values above if you only "trade in radians" and simply say "we do not accept your ancient degree-coins in my shop anymore".


--you will most likely only need only these 2 functions, often, so just eat them!---

getting that out of my system, i noticed that only 2 functions keep being important in trigonometry EVER (that can not be solved geometrically like i prfer to think about things)

https://en.wikipedia.org/wiki/Law_of_cosines (almost exclusively to use ArcCos to get an angle when you only know 2 lengths of a right triangle.)
and
https://en.wikipedia.org/wiki/Dot_product (almost exclusively to calculate distances of planes in 3d space with normal vectors)
and the 2 are closely related anyways.

you keep needing these to calculate with normal vectors and projections, which is done nonstop in any 3d projection, 3d rendering...

--- quaternion rotating in my grave---

the same applies with quaternions, whne it comes to rotations, that applies to  degrees (when it comes to angles that you rotate by (around an axis)), do NOT use Euler-notation for rotations, use quaternions-rotation instead. while you can concert the 2, its much more tricky than a simple conversion-factor (and yes, the very same factor is also included in it)

at one point you will realize that quaternions are a famous, useful and very complex thing, literally full of complex/imaginary numbers, VERY hard to truly understand its mathematical inner workings BUT easy to visualize and therefore easy to use (at least i think) with the abstraction below:

it doesn't even matter if you understand how the magic of quaternion rotation actually works, because the same function can always be computed with less and more accuracy/time/memory use, and thers always multiple different ways to do the same. you rarely calculate quaternion rotation with a pen and sheet of paper, its done by computers, and you just use simple functions that you dont need to know how they work, you just must know how to (NOT) use them!

think of quaternions as describing the position and rotation on a ship on the ocean.

the position of the ship is 2 coordinates on the sphere, latitude and longitude.
but the ship also has a looking-direction on the spheres surface.
imagine it is looking at another shop on the same sphere, that also has a position set by latitude and longiture.

there, you have 2 latitudes and longitudes to determine the position and viewing diredtion of a ship on the surface of a sphere.

quaternions are VERY much like that.

they do not "drid lock" like euler-rotations, and are easyer to deal with in computer code.

if all else fails when it comes to quaternion rotations, just thin in the https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representation of a rotation. its as easiy as holdign a single pen and rotating it around its own main axis, to visualize a quaternion rotation in axis-angle representation. spinning the pen is like spinnign a ship on the tip of it, with the pens length being the diameter of the earth/ocean-surface.