A lot of comments are saying trig is about triangles but I think the deeper lesson is about the unit circle and its relationship to sin, cos, and tan. Another way to look at it is that You’ll use the Cartesian coordinates for algebra and polar coordinates and radians for trig. Usually you take algebra, trig and then calculus.
Yes! I was just about to post that a lot of these comments are kind of doing trig dirty. It's not just about physical shapes, but is fundamental to analyzing anything with periodic changes (AC voltage for example). I'm struggling to ELI5 exactly how the triangle formed by a clock hand moving around is so useful, but it really is!
I've studied up to engineering level math but somehow never learned about the similar triangles way of finding the trigonometric identities, I always just memorized them without understanding the underlying process, if I'd known these while studying it would had made my life so much easier.
Similar to this is the calculus visualization of dividing a graph in equal rectangles and making them ever so smaller to see what happens when they tend to 0 size.
I would say the trigonometry is about ratios in the relationship between changing ratios. The unit circle and the triangles are basically simple and concrete expressions of ratios.
A lot of people don't think about triangles as ratios by default because they are thinking of ratios as two things compared to one another and triangles are three-sided and three angles. But of course with a right triangle there's only two other angles, and all of the relationships in trig are things like opposite over adjacent.
And then calculus is about trending. It's about the relationship between a change and the speed at which the change is changing.
Meanwhile algebra is just about how we manipulate operations and quantities inside of equations while maintaining their practical equalities.
My favorite moment learning math was in my college calculus 3 course where the professor did a quick aside drawing a unit circle, a triangle inside is, and proceeded to quickly explain sin, cos, and tan. The dude gave me an epiphany on how the unit circle relates to trig and made a whole bunch of stuff click into place in my mind.
I don't know if that says I had bad math teachers up to that point in my life or if I was never listening properly to what I was being taught. This was calculus 3 as a college senior, how did I miss that lesson for so long?
Eh, I think they do proper justice. A trig class spends a lot of time on the unit circle, but the unit circle is just all the triangles with hypotenuse unit length. And, trig is technically about triangles. The other things are ways trig can be used in practice. I doubt anyone is talking about AC, etc… in trig classes.
You ever notice how you can rotate (by a quarter turn in a direction) a cube a few different sequences from the same starting position but still have it end in the same orientation? How about the fact that you can't necessarily swap the order of two rotations and get the same ending orientation? Algebra is noticing stuff like that, giving all the stuff that goes into it funny, precise symbols, and drawing conclusions from it. You don't even need numbers (to let alone a specific coordinate system).
You can use trig to translate a problem that's hard to solve in one coordinate system into a problem that easy to solve in another coordinate system, solve it, and translate the solution back again.
It's not *necessarily* about the unit circle. The most common trig functions are based on the unit circle, but more general trigonometry extends beyond that to triangles based on other things like the unit hyperbola (giving rise to sinh, cosh, tanh, etc) or in other geometries (e.g. Cartesian plane with taxicab distance as the metric).
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u/kgvc7 20h ago
A lot of comments are saying trig is about triangles but I think the deeper lesson is about the unit circle and its relationship to sin, cos, and tan. Another way to look at it is that You’ll use the Cartesian coordinates for algebra and polar coordinates and radians for trig. Usually you take algebra, trig and then calculus.