Trig identities

1. Mar 11, 2005

nelraheb

Trig identities plz help

In triangle ABC if sin (A/2) sin (B/2) sin (C/2) = 1/8
prove that the triangle is equilateral plz show steps

2. Mar 11, 2005

James R

Perhaps you can use the sine rule (?)

3. Mar 12, 2005

nelraheb

Sure .. I tried but had no success .. If you find an answer plz post your steps

4. Mar 12, 2005

A_I_

one method suggested:

(an absurd reasonning)

if it is an equilateral triangle then:
A = B = C = pi/3 rad

implies ---> A/2 = B/2 = C/2 = pi/6 rad

implies ---> sin(A/2) = sin(B/2) = sin(C/2) = 1/2

implies ---> sin(A/2)sin(B/2)sin(C/2) = 1/2*1/2*1/2 = 1/8

thus it is indeed an equilateral triangle

if i come with another one i will post it :)
hope it will help

5. Mar 12, 2005

VietDao29

Try expand the equation
$$\sin{\frac{A}{2}}\sin{\frac{B}{2}}\sin{\frac{C}{2}} = \frac{1}{8}$$
to
$$4\sin{\frac{C}{2}}^{2} - 4\sin{\frac{C}{2}}\cos{\frac{A - B}{2}} + 1 = 0$$
Then to
$$(2\sin{\frac{C}{2}} - \cos{\frac{A - B}{2}})^{2} + (\sin{\frac{A - B}{2}})^{2} = 0$$
Now you have something like $A^{2} + B^{2} = 0$ so
$$\left\{ \begin{array}{c} A = B \\ \sin{\frac{C}{2}} = \frac{1}{2}\cos{\frac{A - B}{2}} \end{array}\right$$
So you will have A = B = C = 60 degrees, which implies the triangle ABC is equilateral.
Hope it help.
Viet Dao,

6. Mar 12, 2005

nelraheb

For AI thank you but this won't do

7. Mar 12, 2005

nelraheb

For VietDao29 If A^2 + B^2 = 0 then we're stuck because no two +ve numbers addto zero ...right ? Then it should be A^2 = - B^2
How did you expand 1st step
How did you get last step
plz go in more details

8. Mar 12, 2005

dextercioby

Both A and B are real.So their square is larger or equal to zero.In order for the sum of the squares to be 0,each if the squares must be 0.

Daniel.

9. Mar 12, 2005

nelraheb

Well that's a good point. How did I miss that :) Now for the first step plz how did we expand Sin (A/2) Sin (B/2) Sin (C/2) to next step
ie. How to start .... the rest is ok

10. Mar 12, 2005

dextercioby

Use this IDENTITY:

$$\sin x\sin y\equiv \frac{1}{2}[\cos(x-y)-\cos(x+y)]$$

The result is immediate.

Daniel.

11. Mar 12, 2005

Tom Mattson

Staff Emeritus
Try starting by eliminating a variable. Since you know that A, B, and C are all in the same triangle, you have:

A+B+C=180
C=180-A-B

See where that gets you.

12. Mar 12, 2005

nelraheb

Thank you all ... I can do it now following your steps
The rule supplied by Dextercioby did not look familiar (but it's correct I checked) ..well memory is not what it used to be :) isn't that a bit complicated though ... I thought the answer should be more straight forward .. any way thank you all again

13. Mar 12, 2005

dextercioby

That's interesting.The checking part.I've said IDENTITY. There may have been a chance i didn't invent it,but either picked it from a book or deduced starting other identities (which i have actually done).

Daniel.