Math Is Fun Forum

fusilli_jerry89

Member

Registered: 2006-06-23

Posts: 86

180 degree triangle?

I was doin this physics question ,adding vectors and such and got the right answer. What didnt make sense was that the angles of the triangle did not add up to 180 degrees.

I had a triangle with 3 sides (70.7, 40, 33.7) with an angle of 15 degrees between the 70.7 and the 40. The problem is, when I use sine law to figure out the remaining 2 angles, they do not add up to 180 degrees. What is going on here?

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: 180 degree triangle?

Let me try without knowing about the 15 degrees part at all.
180 = angleAcrossFrom70.7 + angleAcrossFrom40 + angleAcrossFrom33.7
180 = aAF707 + aAF40 + aAF337
70.7 = constant_k times sine(aAF707)
40 = constant_k times sine(aAF40)
33.7 = constant_k times sine(aAF337)

---

Because 70.7 is 3 less than 40 plus 33.7, then angleAcrossFrom70.7 is about 175 degrees.
This is important, because now you know you are in the 2nd quadrant for that angle.
And the other 2 angles are small in the first quadrant.

---

For small angles, the sine is almost linear, so we can estimate with equation:
Just kidding, but nice idea.
Let's see, the height of a curving circle is the sine... along the very vertical part between 0 and 10 degrees and 175 to 180 degrees or so.
One guess is 180 - 7, 3 and 4 degrees for the angles. So 173, 3, and 4 degrees is just a guesstimate.

---

Now the 3 that is off on the sum of the 40 and 33.7, is 10% or so of the smallest side, and 4% of the longest side.
So the curve of the circle accounts for this discrepancy somehow.  Else it would add up exactly to 70.7 instead of 73.7.

---

Maybe we should also learn the "law of cosines", as this added information might help.
I was using the law of sines proportions in above thoughts.

---

One other thing.  I just realized that a circle is almost linear at microscopic inspection.
Hence if you have 3 or more data angles in very close proximity, the sine differences should be approximately
proportional to the small angle differences. 
This is one reason why the sine of small radian angles is the radian value, but there is another reason too which I don't know.

---

So the 3 angles are about 15, 18, and 147, I just noticed with a calculator if I do
use your 15 to start off with.
Just less than 18, and just about 147.1 I got.

---

147.1131053 says my calculator.
17.89078786
and 15 degrees was given, hopefully correct.
Let's add'm up and see what it is...
They add up to 180.0038932
I guess the 15 degrees given is simply not correct.
That would be my guess...

---

Or if the 15 degrees is right, and the 40 is correct, then I tried 70.667 and 33.667 and I got 180.0030171, a little closer to 180.

---

Now I will try 13 degrees and 40 and 70.7 and 33.7 and see what I get...
This yields 180.3264306 if I pressed the right buttons.
That is way worse off...

---

Now I will try 5 degrees and 40 and 70.7 and 33.7 and see total angles...
This yields 180.4022347, which is even further off.

---

Now I will try 3 degrees since that was my original guesstimate, now I think I was way off.
3 degrees, 40, 70.7, 33.7 for sides, 3 degrees is opposite 33.7 side.
This yields 180.2578951.  Good Lord, it is closer than the 5 degrees one!!!!  WEIRD ~?~?~

---

Now I'll try 0.25 degrees for the fun of it, just to see what it does.
0.25 degrees across from 33.7 side, and also 70.7 and 40 sides.
Well, if I pushed the right buttons, this imaginary example reveals 180.0222499.
These are all funny-math calculations because we are supposing an object
that doesn't exist, to see what happens.

---

I just learned the "law of cosines" on a web site and the 15 degree angle came out to 15.0186377 degrees.
Provided I pressed the right buttons again...

---

This now makes the other angles 17.91324219 and 147.0681201.
Hope that helps!! (smile)

Edited to combine lots of posts into one and make everything look neater and stuff.

Last edited by mathsyperson (2006-12-04 06:08:18)

---

igloo myrtilles fourmis

Ross Barker

Guest

Re: 180 degree triangle?

you sick, sick people

Devantè

Real Member

Registered: 2006-07-14

Posts: 6,400

Re: 180 degree triangle?

We're mathematical geniuses is all.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: 180 degree triangle?

This reminds me that I want a triangle solver on the website, not just the solution but the steps.

Could make it too easy for homework, though!

---

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman