Triangle ABC is such that AC=BC and AB/BC = r

Show that cos A + cos B + cos C = 1 + r -(r^2/2)

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- Jun 11th 2007, 01:14 PMPh4mChallenging Trigonometry Question
Triangle ABC is such that AC=BC and AB/BC = r

Show that cos A + cos B + cos C = 1 + r -(r^2/2) - Jun 11th 2007, 01:50 PMJhevon
My proof is a bit long winded. i'm sure someone else will come up with a better one soon.

See the diagram below:

what was described was an isoseles triangle, so we can further assume that:

(1) AD = BD

(2)

Now from the triangle, we see that:

also,

also,

Further more, you should know the identity,

Now to start peicing this together:

Since ,

Also, since ,

and now we're done. just to say what we have to say:

as desired - Jun 11th 2007, 02:23 PMThePerfectHacker
- Jun 11th 2007, 02:26 PMJhevon
- Jun 11th 2007, 02:36 PMThePerfectHacker
- Jun 11th 2007, 02:41 PMJhevon
- Jun 11th 2007, 04:08 PMSoroban
Hello, Ph4m

I'll finish what ThePerfectHacker started . . .

We have isosceles triangle ABC with and

Then the triangle looks like this:Code:`C`

*

/ \

/ \

a/ \a

/ \

/ \

A * - - - - - * B

ar

From the Law of Cosines, we have:

. .

. .

. .

Therefore: .