If your GC, who should know pretty well how you can handle the courseload, is recommending taking PreCalc, you should probably listen to their advice. As others have mentioned, you should confirm whether he or she will mark your courses as the most rigorous, because if your classmates get that and you don’t, that could be an issue, but jeopardizing chances to any college is tough to say right now. Having Calc down the road and taking honors in science classes will help.
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Do you have an EC where it would be more valuable to spend time and energy than a course that makes you feel overwhelmed and no doubt takes hours a day?
I cannot answer your question as to the effect on your chances of admission to Columbia, but this situation may be a bit of confirmation that your intention & desire to major in history and/or law is probably the better path for you. Most of the lawyers with whom I am familiar are unacquainted with higher level math.
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Pre-calc as a sophomore is still pretty accelerated so I don’t see any issue with it. At many HS’s getting to Calc as a senior is the goal for students on the most rigorous path. This student is only a sophomore and would, presumably, be taking AP Calc (of some sort) as a junior. For some kids it takes more time to absorb the material - better to have an excellent grounding in pre-calc than to advance and not really internalize the material.
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One way to think about this is what if another kid from your school applied to Columbia? How can you make your application stand apart from theirs? This is not meant to compete with them, but to highlight who YOU are in your class choices, your extra curriculars, etc. Let it all tell a story about your strengths. Make it hard for them to turn you down.
You also have plenty of time to research other schools which realistically even the the best kid, at the best high school, who got the A in Math Analysis will need to do.
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Don’t obsess over it. It’s not healthy. Just get the best grades you can. If you get in, GREAT. If you don’t, then there’s plenty of options available.
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