Proving an Equilateral Triangle: Trig Identities Explained

In summary, the conversation discusses a problem involving trigonometric identities. The goal is to prove that a triangle is equilateral using the given equation and steps provided by various participants. The conversation covers different methods and identities, with some participants finding success in solving the problem. Despite some initial confusion, the participants eventually reach the conclusion that the triangle is indeed equilateral.
  • #1
nelraheb
6
0
Trig identities please help

In triangle ABC if sin (A/2) sin (B/2) sin (C/2) = 1/8
prove that the triangle is equilateral please show steps
 
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  • #2
Perhaps you can use the sine rule (?)
 
  • #3
Sure .. I tried but had no success .. If you find an answer please post your steps
 
  • #4
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
Try expand the equation
[tex]\sin{\frac{A}{2}}\sin{\frac{B}{2}}\sin{\frac{C}{2}} = \frac{1}{8}[/tex]
to
[tex]4\sin{\frac{C}{2}}^{2} - 4\sin{\frac{C}{2}}\cos{\frac{A - B}{2}} + 1 = 0[/tex]
Then to
[tex](2\sin{\frac{C}{2}} - \cos{\frac{A - B}{2}})^{2} + (\sin{\frac{A - B}{2}})^{2} = 0[/tex]
Now you have something like [itex]A^{2} + B^{2} = 0[/itex] so
[tex]\left\{ \begin{array}{c} A = B \\ \sin{\frac{C}{2}} = \frac{1}{2}\cos{\frac{A - B}{2}} \end{array}\right[/tex]
So you will have A = B = C = 60 degrees, which implies the triangle ABC is equilateral.
Hope it help.
Viet Dao,
 
  • #6
For AI thank you but this won't do
 
  • #7
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
please go in more details
 
  • #8
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
Well that's a good point. How did I miss that :) Now for the first step please 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
Use this IDENTITY:

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

The result is immediate.

Daniel.
 
  • #11
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
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
nelraheb said:
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

That's interesting.The checking part.I've said IDENTITY. :wink: 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.
 

What is an equilateral triangle?

An equilateral triangle is a triangle with all three sides of equal length. This means that all three angles of the triangle are also equal, measuring 60 degrees each.

What are trigonometric identities?

Trigonometric identities are mathematical equations that involve trigonometric functions such as sine, cosine, and tangent. These identities are used to simplify and solve trigonometric equations.

How can trigonometric identities be used to prove an equilateral triangle?

The most commonly used trigonometric identities for proving an equilateral triangle are the Pythagorean identities, which state that sin^2x + cos^2x = 1 and tan^2x + 1 = sec^2x. By using these identities and manipulating the equations, it is possible to show that all three sides and angles of an equilateral triangle are equal.

What other methods can be used to prove an equilateral triangle?

In addition to trigonometric identities, other methods such as congruence and similarity can also be used to prove an equilateral triangle. These methods involve comparing different parts of the triangle and using geometric theorems to show that they are equal.

Why is it important to prove an equilateral triangle?

Proving an equilateral triangle is important because it allows us to confirm the properties of the triangle and use them in further calculations or proofs. It also helps to establish the relationships between the sides and angles of the triangle, which can be useful in solving various geometric problems.

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