Proving the Sum of Cosines in a Triangle Using Half-Angle Formula

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In summary, the conversation discusses a proof for the equation cos(A)+cos(B)+cos(C)=1+4sin(\frac{1}{2}A)sin(\frac{1}{2}B)sin(\frac{1}{2}C) for any triangle ABC. The conversation suggests using product to sum formulas and potentially substituting C with 180° - (A+B) or C =##\pi## - (A+B) to simplify the equation. The person mentions trying to use double angle formulas but is unsure how to proceed. They also suggest doing a Google search for triangle "sin A + sin B + sin C" for further help.
  • #1
ciubba
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Prove for any triangle ABC that:
[tex]cos(A)+cos(B)+cos(C)=1+4sin(\frac{1}{2}A)sin(\frac{1}{2}B)sin(\frac{1}{2}C)[/tex]

I tried using the half-angle formula on the right side to get:

[tex]1+4 \left(\frac{1-cos(A)}{2}\frac{1-cos(b)}{2}sin(\frac{1}{2}C)\right) {}[/tex]

which simplifies to

[tex]1+\bigg(1-cos(A)-cos(B)+cos(A)cos(B) \bigg)*sin(\frac{1}{2}C)[/tex]

by the product to sum rule, this simplifies to

[tex]1+\bigg(1-cos(A)-cos(B)+\frac{1}{2}\Big[cos(A+B)+cos(A-B)\Big]\bigg)*sin(\frac{1}{2}C)[/tex]

I tried expanding the sin 1/2 C, but that just made things even more complicated. How should I approach this problem?
 
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  • #2
Use product to sum formulas.

I remember their being a shortcut for this problem where u rewrite the terms on the left allowing you to use double angle formulas (dont quote me on this). I'll try to give a shot.
 
  • #3
Forget the last part I said about product to sum. It just gives you back the original statement. I'll try to work on it later after work. Just had a 15 min break.
 
  • #4
Don't forget at some stage you are going to substitute C with 180° - (A+B)
or C = ##\pi## - (A+B)

whichever you prefer.
 
  • #5
If stymied, it's often worth trying a google search, such as: triangle "sin A + sin B + sin C"
 

1. What is the definition of triangle ABC equality?

Triangle ABC equality refers to the concept that two triangles, ABC and DEF, are considered equal if all three corresponding sides and angles of the triangles are congruent.

2. How can you prove triangle ABC equality?

Triangle ABC equality can be proven using various geometric theorems and postulates such as the Side-Angle-Side (SAS) or Angle-Side-Angle (ASA) congruence criteria, or the Pythagorean Theorem.

3. What is the importance of proving triangle ABC equality?

Proving triangle ABC equality helps to establish the congruence of two triangles, which is necessary for solving many geometric problems and proving other theorems.

4. Can you prove triangle ABC equality without using congruence criteria?

No, proving triangle ABC equality requires the use of congruence criteria or other geometric theorems and postulates. Without these, it is not possible to establish the congruence of two triangles.

5. How does proving triangle ABC equality relate to real-world applications?

Proving triangle ABC equality is important in fields such as engineering, architecture, and surveying, where precise measurements and constructions of triangles are necessary for designing and building structures.

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