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"You should be able to prove this"

Anonymous Thu 29 Jul 10:43:26 2021 No.13448104 View Reply Original Report

Quoted By:

Prove that:
a(1) < b(1)
a(2) < b(2)
a(3) < b(3)
I was reading a calculus textbook and the author says it is a simple matter to prove this using elementary geometry.

Anonymous

Anonymous Thu 29 Jul 2021 11:33:34 No.13448211 Report

Quoted By: >>13448215 >>13448303 >>13448358

It should be a trivial observation that increasing or constant the lengths of any 2 sides of a triangle while keeping an angle constant will require that the length opposite the fixed angle increase - this is basically what the sine rule says. The cosine rule also works, eg for a1 and b1:

Let R be the radius of the circle and x be the increase in length of the first line from the bottom (the upper side of the lowest triangle):

a12= R2 + R2 - 2R2cosA1 Using cosine rule
b12= R2 + (R+x)2 - 2R(R+x)cosA1
= (2R2-2R2cosA1) + 2Rx + x2 -2RxcosA1
= a12 + x2 + (2Rx - 2RxcosA1)
? a12 + x2 Since cosine of an acute angle is less than 1
b1 ? a1

Anonymous

Anonymous Thu 29 Jul 2021 11:34:56 No.13448215 Report

Quoted By:

>>13448211
The 2's should all be squares.

Anonymous

Anonymous Thu 29 Jul 2021 12:19:21 No.13448303 Report

Quoted By: >>13448358 >>13448471

>>13448211
elementary geometry $\neq$ trigonometry

Anonymous

Anonymous Thu 29 Jul 2021 12:46:03 No.13448358 Report

Quoted By:

>>13448211
Thanks.
>>13448303
The book I'm reading explains the law of cosines just a few pages before it uses the picture in the OP to prove the limit (sin x)/x as x tends to zero, so i guess it's fine to invoke trigonometry to prove those inequalities.

Anonymous

Anonymous Thu 29 Jul 2021 13:25:33 No.13448471 Report

Quoted By: >>13448550

>>13448303
Trigonometry is a subset of geometry. I would consider sine/consine rule to be elementary geometry.

Anonymous

Anonymous Thu 29 Jul 2021 13:52:04 No.13448550 Report

Quoted By:

>>13448471
but you don't need to invoke trigonometry, reasoning about areas is enough

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