All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Acute triangle, Trigonometric function! (Posted on 2007-10-13)
Prove that in an acute triangle sin(A) + sin(B) + sin(C) > 2

 See The Solution Submitted by Chesca Ciprian Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 re: not a proof but ... | Comment 2 of 9 |
(In reply to not a proof but ... by Charlie)

It's easy to show the result for right triangles:

In a right triangle, the sine of the 90-degree angle is 1.  The other two angles are acute and complementary. Each one is greater than zero degrees.  If one is called x, the sum of the sines of these other two angles is sin x + sin(90-x). That this total always exceeds 1 is illustrated by the fact that if the portion of the sine curve between zero and 90 degrees were changed into a straight line connecting (0,0) to (90,1) -- considering the function in degree measure -- the straight line would represent the condition under which sin x + sin(90-x) would be exactly 1. But the true sine curve is always above this straight line, so sin x + sin(90-x) always exceeds 1, except at the end points when one is zero and the other 90.

I tried to use this by dividing the acute triangle into two right triangles. The sum, then greater than 4 would be diminished by the 2 for the right angles lost, but I couldn't account for the two acute angles that would merge into one, as the sine of the combined angle would be less than the total of the sines of the two angles when they were in separate right triangles.

But combined with the table in the previous post, the totals look higher for acute triangles than for right triangles.  But how to prove it....?

 Posted by Charlie on 2007-10-13 16:47:30
Please log in:
 Login: Password: Remember me: Sign up! | Forgot password

 Search: Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2018 by Animus Pactum Consulting. All rights reserved. Privacy Information