Jomo enjoys beautiful solutions

Steven G

Steven G

Elite Member
Joined
Dec 30, 2014
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JeffM said:
Let [MATH]u = sin(x) \text { and } sin^2(x) + \dfrac{1}{sin^2(x)} = sin(x) \implies[/MATH]
[MATH]0 < |\ u \ | \le 1 \text { and } u^2 + \dfrac{1}{u^2} = u \implies[/MATH]
[MATH]u^4 - u^3 + 1 = 0.[/MATH]
[MATH]0 < u^4 \le 1 \implies 1 < u^4 + 1 \le 2.[/MATH]
[MATH]- 1 \le - u^3 \le 1. [/MATH]
From that we conclude what?
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Very nicely done, just like a true algebraic would have solved it.
 
Jomo said:
Very nicely done, just like a true algebraic would have solved it.
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Yes, on both counts. Very nice. Yet, it's not what a tutor would have done. (I wish Jeff had left more for trivun to work out.)

I'd still like to see the original exercise statement, but, regardless of what's asked when assigning that equation, I think I'll continue my first post later because Jeff's kind response led me to see I was too terse; I don't think I got my point across very well.

Sometimes, it's good to remember what we learned in grade school about proper fractions and their reciprocals (improper fractions). Sometimes, it's productive also to consider meaning, when looking at each side of a given equation, before trying to solve. I like to encourage trigonometry students to keep basic graphs in mind, when they get stuck solving simpler equations. Maybe they get an idea, from that.

?
 
Otis said:
Yes, on both counts. Very nice. Yet, it's not what a tutor would have done. (I wish Jeff had left more for trivun to work out.)

I'd still like to see the original exercise statement, but, regardless of what's asked when assigning that equation, I think I'll continue my first post later because Jeff's kind response led me to review. My reply was too terse, for a broader audience.

Sometimes, it's good to remember what we learned in grade school about proper fractions and their reciprocals (improper fractions). Sometimes, it's productive also to consider meaning, when looking at each side of a given equation, before trying to solve. I like to encourage trigonometry students to keep basic graphs in mind, when they get stuck solving simpler equations. Maybe they get an idea, from that.

?
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I guess that I spoke too quickly. Yes, maybe that was not the best way to show a weak student but not so bad for a strong student. My response was just based on how he solved the problem. Truly an elegant solution.
 
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@Otis

It’s always a struggle to figure out how much to say. I’m sure most of us have sometimes said too much and sometimes said too little. In this case, trivun clarified in post 3 what the question really was and knew the answer to that question from post 2. Consequently, I felt that it was important that tivun know more than just the answer so perhaps I went too far.
 
JeffM said:
… trivun clarified in post 3 what the question really was …
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That's a place where we differ.

JeffM said:
… I felt that it was important that [trivun] know more than just the answer …
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Hard to know at this point what answer is expected. Perhaps, they were asked to provide some explanation.

I think trivun had believed there were solutions, but how can we tell from, "Could anyone help me solve this one?"

I don't find that it's always a struggle to figure out how far to go. When I can't see what a student has tried or thought about, I mostly struggle with deciding what kind of approach to take for offering hints or suggestions or examples.

?
 
Enough! Regardless of whether or not Jeff's solution was appropriate for the student, the solution was beautiful. We can't overlook that.
 
As someone who studied quite a bite of algebra in graduate school and dreamed about being a research algebraist it is true that I like to see beautiful algebraic (and other branches of math) solutions.

Here is the most beautiful proof that I saw in a while. Truly elegant does not do justice to this proof. And it is not in the field of algebra.
 
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