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I feel bad I keep having to reference my notes throughout my TRIG class

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Physics PhD here. I relied on my notes throughout my studies, even at PhD stage (it gets better, with simpler stuff being committed to memory as you use it more and more, and what you need your notes for becomes increasingly complex). The point of learning science isn’t to memorise things, it’s to learn to think analytically. It doesn’t matter if you can’t remember a formula (some trig formulas I still don’t quite remember), it matters that you develop an intuition so you can use that formula to help you derive an interesting result.
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Hello, I am a PhD student in math and I would be absolutely helpless without my notes or my small collection of textbooks. You are not alone. Even the best mathematicians (which I certainly don't claim to be) are constantly referring to their notes. You are not alone.

On a more philosophical note, what is the purpose of note-taking if not to read your notes at a later time? We're only human, our memories are limited, and notebooks are nothing more than external hard drives that help us recall the information we already learned.
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Just think of notes as extension of your own memory. Slower to access but much more reliable for storage.

Anything you need to access regularly, you should move them to your brain which is faster. But it's fine to not remember things that don't get used as often. The process is generally automatic, as you have to keep recall one piece of information again and again, your brain just automatically remember them.

If there are things you use regularly but somehow you can't move to your brain, maybe some focused practice could help? Or maybe if you have an exam coming up when you specifically cannot use notes and they might ask about something you don't use regularly. It also helps to make use of your intuition.
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Depends on the equation, honestly. Something like sin² + cos² = 1, should be second nature. Embarrassingly, the sin = opp/hyp is something I always only remember by reciting sohcahtoa. And then you got stuff like sin(x+y) = ... which I just google every single time.

So no, as a mathematician I don't remember many trig equations. Google exists and I make use of that. What matters is not that you remember the equations but that you know they exist and what they mean. If I see for example an integral of cos²(x), I know I need a formula to express this in terms of cos(2x), but I don't remember what formula that is, I just google that. And of course I should know what sine means exactly both in triangles and the unit circle.  


It's never a good idea to memorize in math. Rather, you should remember something by using it a lot or by "visualizing" what is going on. I know that cos² + sin² = 1 since I use it every single day. And I know that sin(pi - x) = sin(x) since I just see the unit circle in front of me which makes this relation obvious, I haven't memorized this, nor can I recite it from memory.
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That's what notes are for. My students keep not referring to their notes when doing homework. One or two problems per written assignment come directly from the notes or textbook with only modest modification as a means to force them to look at both. They know this, and still they refuse to look at their notes. It drives me crazy.
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Just to add to the good responses here. I'm a math teacher, precalculus and calculus, and I allow my students to use formula sheets for things like sum/difference, double angles, etc. Same with calculus and some of the integration formulas.

But, I'd caution that not knowing basic trig relationships can slow you down and distract you from the problem at hand. Definitions of the trig functions, sohcahtoa or whatever you use, ratios like tan =sin/cos, etc. Your problem solving will be much smoother and efficient if you don't need to be constantly looking these up.

I find that students who often look away from their work to find a relationship on a reference sometimes lose track of their thought process and waste time backing up to move forward again.
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I'm a theoretical physicist (finished my PhD awhile ago) doing stuff that often involves a fair bit of trig. Whenever I'm doing that stuff I keep a tab open of trig identities.
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Undergrad here. I look at my notes literally all the time. Some things you ingrain into your head through familiarity and repeat usage, but other things you simply don’t use enough to memorize. That’s what notes are for.
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Don't ever feel bad for having to check notes!  You are not a computer.  Your brain should be trained to find the correct answer. The path there doesn't matter all that much. In college Stats our teacher let us have ALL the formulas. It's on you to understand how they work together and how to get understand the question well enough to use the formulas to get to the write answer.  You're not expected to memorize every formula in math, just like you don't need to know specific dates in history, just how to get to the thing you're looking for efficiently.
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i only remember half of the things, and the other things i just remember how tp re-discover it every time i need to use it

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