Hey, you know what gets used a lot in chemistry?
Math.
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Approximately two minutes.
Pretty much the same technique as everyone else:
simplify (sec x - tan x)**2:
(1/cos x - sin x / cos x)² = (1 - sin x)/(1 + sin x)
((1 - sin x) / cos x)² = (1 - sin x)/(1 + sin x)
(1 - sin x)² / cos² x = (1 - sin x)/(1 + sin x)
Cross multiply:
cos² x (1 - sin x) = (1 + sin x)(1 - sin x)²
Factor (1- sin x) out of both sides, after observing that sin x = 1 is a solution to the original problem:
cos² x = (1 + sin x)(1 - sin x)
Multiply out the right-hand side:
cos² x = 1 - sin² x
Add sin² x to both sides and get left with Pythagoras:
cos² x + sin² x = 1
To present a proof of the original formula, given the identities stated, start from Pythagoras and reverse these steps.
It took me at least five times as long to type this as to solve the problem.
I got it in about 5 min, but my buddy ‘Photomath’ did it in about .1 seconds: just point the camera at the equation and press start…
I love that app 
It not only solves the whole problem, it also tells you step by step how it did it, which techniques it uses etc.
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And a ton of thermodynamics in learning chemistry, yet thermo is upper-division physics. Might take a long time getting to freshman chemistry that way.
Putting together a curriculum is not trivial. I know someone – an utterly brilliant professor of physics – who created a 2-semester introductory course for physics majors based on de Broglie’s lecture of starting with waves. Good course, great start for physics majors, but still turned out to require a more traditional intro to keep from losing them in the dust. I still recommend the books though (Raymond, A Radically Modern Approach to Introductory Physics) The hope was to shorten the prerequisite sequence enough to cover more than the bare minimum in four years.
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Took me about 30 seconds to do it in my head.
I tell my students (physics majors in college) that the only trig identities they should memorize are tan = sin/cos and sin^2 + cos^2 = 1. The latter one should eventually come to seem intuitively obvious if you really understand what’s going on with trig (particularly if you visualize the unit circle); the former is a definition, and kind of gratuitous except sometimes you see tan. You can look up any other ones you might need (including the definition of sec).
The definition of tan, sec, and that one identity is all you need to solve this one.
(Addendum: there’s one more I tell them to know, and that’s e^(ix) = cos x + i sin x. That one comes up a lot in quantum mechanics.)
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Is it ever. Trying to pull stuff out of the brain from 30 years ago can be hit or miss. I usually run it through WolframAlpha even if I get it - just to make sure if I am offering help with a stubborn problem I am offering correct help.
Identiries are exacttly how you get things done in math. What is wrong with you??
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This is one of those problems where the answer spells a dirty word, right?
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It took me about a minute once I found a pencil, without using anything except the right-angle-triangle identities; using o, a and h to mean opposite, adjacent, hypotenuse, you can rewrite the question, knowing x is constant –
show that
(h/a - o/a)^2 == (1 - o/h) / (1 + o/h)
(h - o)^2 / a^2 == (h - o) / (h + o)
(h - o) (h + o) == a^2
h^2 == a^2 + o^2
which is true for a right-angled triangle
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Thank you. That was just the right amount of challenge. Hard enough to be a few minutes challenge and for me to smile when I got it, easy enough that I don’t feel despondent at my skill loss.
Sort of like the Thursday NYT Crossword Puzzle.
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Actually, a great number of highschools have already become “physics first,” which generally means physics then chemistry then biology, precisely because they build on each other.
One difficulty is that calculus and trig are used so much in physics, so it can be hard to do physics freshman or sophomore year before learning the math. You can do physics without calculus, of course, and many schools do it well, but physics and calc go so well together it’s a pity. Some schools therefore do chem -> bio -> physics.
It’s not easy balancing all the different concerns. But there are a lot of people who think seriously and with lots of care about these issues, and it’s annoying to hear all curriculum design dismissed as if it’s by idiots who haven’t learned anything in a hundred years.
(Another alternative, of course, is to use the British system of doing all the sciences throughout high school. That’s the way I was taught and whenever I ask why Americans don’t do that I’m just told about “block scheduling,” which doesn’t really make sense to me.)
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Foil it, then use a conjugate? That’s what I would try.
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Let’s check with Katherine Johnson:
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Thanks for the memory, I really liked that movie!
Euler’s formula is a nice gateway to the Fourier transform, if that turns out to be useful in the future.
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I literally did it in my head inside of a minute prior to looking at the comments.
Wind back to 1983 and my introduction to basic trig, I struggled to remember basic things like sin(30) = 1/2, cos(45) = 1/sqrt(2), etc. My friend happened to have this bizarre portable tape recorder that let you loop between two points. So I recorded myself speaking the basic trig tables, put it on loop, and went to bed hoping my subliminal mind would learn. Didn’t work and I failed the test. I had totally missed the point. Those tables are trivial to derive if you understand how they arise. That is, if you have a geometric understanding of what’s happening. No doubt our teacher showed us this … but I was probably stoned, not paying attention, or both. And so I fixated on rote memory over understanding.
I guess you also hate programming courses, because the example problems deal with “widgets” and are not directly implementable in financial cost models.
I love math.
Really? I sense that you and math had a bad break-up.
Why is this being taught instead of probability? Statistics? Decision making? Even symbolic logic?
You’re acting like she was being taught how to use logarithm tables. This is just straight geometry and it’s damned useful.
I hate this question. I hate what the system is doing to students like your daughter.
What is it doing to her, besides helping her to think more creatively. What is the point of a jigsaw puzzle? "Gee, all the pieces are all jumbled and of irregular shape! If you had wanted me to assemble this, why arent the pieces numbered, in sequential order?!"
My guess is that the purpose of this question is to ensure compliance and screen for persistence.
You’re overthinking it. Sit down and take a deep breath. This math puzzle was not state propaganda.
The part I disagree with is dressing it up with words (secant?) and extraneous concepts that simply hide the fun part of the puzzle.
“extraneous concepts”? if you ever look into the history of math, you’ll find that what we learn is a much streamlined version of how awkward math has been. plus, you act like someone willfully invented “secant” as an impediment to a solution.
Trigonometry is a miracle. But it’s not a miracle that belongs in tenth grade. Problem solving is essential, and yet it’s omitted (by most curricula) except when necessary for testing purposes.
So, we keep trigonometry out of reach of these innocents until they reach the age of consent cosecant?
It makes me sad.
If your post had been in jest, I would be giving you a standing ovation.
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You may be an expert in your field, but please don’t judge too easily about other subjects. (Remember the Dunning-Kruger effect). As an engineer, I very often encounter the type of problems that require this kind of algebraic skills in my work, so to me it seems useful to teach this to kids. Now the US may have outsourced all engineering to Asia, but if you ever want to start building stuff again, you need people who can do this kind of problem solving. And yes, they also need probability and statistics and a lot of other skills.
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I always find discussions like this interesting because it shows me people who genuinely love mathematics and related fields and I have no idea how they get that love.
I’ve had teachers who clearly loved the subject but were utterly unable to inculcate the same in their students.
This is the kind of thing I mean:
Why would I want to? In my experience, people who like maths would never think to provide that information. It’s not a question that even occurs to them.
Why? I mean I suppose I can see it’s an elegant solution to a philosophical question but other than that?
