tingle

jayz · 2007-03-07 00:25:03

whats the type of the tringle abc if you know that :
cos a . cos b . cos c = 1 / 8

mathsyperson · 2007-03-07 01:36:35

Interesting question.
Well, the cos function is positive for angles strictly between 0° and 90°, it's 0 at 90°, and it's negative for angles strictly between 90 and 180.

Multiplying the cosines of all the angles in that triangle gives a positive number, so that means that none of the cosines can be 0. Also, either 0 or 2 of them are negative. However, it's impossible for there to be 2 angles in a triangle that are both more than 90° (because then the total couldn't be 180°), which means that all of the cosines must be positive.

Therefore, they are all less than 90° and so the triangle is acute.

It's almost certainly possible to get a bit more information if you think about it more deeply, but we've found the type of the triangle, and that's all we needed to do.

JaneFairfax · 2007-03-07 03:22:17

An equilateral triangle is one possibility. But there might be others, of course.

mathsyperson · 2007-03-07 03:44:37

Ooh, that's interesting. I think that might actually be the only possibility then.
I think that the equilateral triangle gives the maximum possible value when you multiply all of its cosines together, and if that's the case, then there won't be any other triangles that fit because all of their answers will be too low.

Stanley_Marsh · 2007-03-07 11:00:37

cosa=cosb=cosc=1/2 my first impression. lol

Stanley_Marsh · 2007-03-07 11:16:51

simplified ,get

Looks very symmetric~lol

Last edited by Stanley_Marsh (2007-03-07 11:18:42)

Math Is Fun Forum

#1 2007-03-07 00:25:03

tingle

#2 2007-03-07 01:36:35

Re: tingle

#3 2007-03-07 03:22:17

Re: tingle

#4 2007-03-07 03:44:37

Re: tingle

#5 2007-03-07 11:00:37

Re: tingle

#6 2007-03-07 11:16:51

Re: tingle

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