Trig trig trig

  • Thread starter mohlam12
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  • #1
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hey everyone,
i have to show that in a triangle, there is : (A, B, C are the angles of that triangle)
[tex] \sin \left( 1/2\,B \right) \cos \left( 1/2\,C \right) +\sin \left( 1/2

\,C \right) \cos \left( 1/2\,B \right) =\cos \left( 1/2\,A \right) [/tex]

for this one, here is what i got to...

[tex] 1/4\,\sqrt {2-2\,{\it cosB}}\sqrt {2+2\,{\it cosC}}+1/4\,\sqrt {2-2\,{

\it cosC}}\sqrt {2+2\,{\it cosB}} [/tex]


my question is, i dont have any A in this equation above, and i have to prove that it is equal to cos(a/2)!! i know that cos(a)=cos(pi-(b+c) ... please if someone can help me with that!

for the second one, we have : [tex] \left( \cos \right) \,\alpha={\frac {a}{b+c}} [/tex]

and [tex] \left( \cos \right) \,\beta={\frac {b}{c+a}}
and \left( \cos \right) \,\gamma={\frac {c}{a+b}} [/tex]

and we have to show that:
[tex] 1/2\,{\tan}^{2}\alpha+1/2\,{\tan}^{2}\beta+1/2\,{\tan}^{2}\gamma = 1 [/tex]

this one, i erally dont know what to do! if someone can help me out! or maybe give me hints...

i appreciate your help!!
 

Answers and Replies

  • #2
TD
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mohlam12 said:
this one, i erally dont know what to do! if someone can help me out! or maybe give me hints...

i appreciate your help!!
You can use the fact that

[tex]\tan ^2 \theta = \frac{{\sin ^2 \theta }}
{{\cos ^2 \theta }} = \frac{{1 - \cos ^2 \theta }}
{{\cos ^2 \theta }} = \frac{1}
{{\cos ^2 \theta }} - 1[/tex]
 
  • #3
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Sorry... for the second one, there is a mistake (I am still learning how to us LaTex so, yeah) here is the correct form: I am not gonna use LaTex, I can't get what I want...

tan²(α/2) + tan²(β/2) + tan²(γ/2) = 1

Thanks,
 
  • #4
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mohlam12 said:
Sorry... for the second one, there is a mistake (I am still learning how to us LaTex so, yeah) here is the correct form: I am not gonna use LaTex, I can't get what I want...

tan²(α/2) + tan²(β/2) + tan²(γ/2) = 1

Thanks,
Then you can use the fact that:

[tex]\tan \left( {\frac{\theta }
{2}} \right) = \frac{{\sin \theta }}
{{1 - \cos \theta }} \Leftrightarrow \tan ^2 \left( {\frac{\theta }
{2}} \right) = \frac{{\sin ^2 \theta }}
{{\left( {1 - \cos \theta } \right)^2 }} = \frac{{1 - \cos ^2 \theta }}
{{\left( {1 - \cos \theta } \right)^2 }} = \frac{{\left( {1 - \cos \theta } \right)\left( {1 + \cos \theta } \right)}}
{{\left( {1 - \cos \theta } \right)^2 }} = \frac{{1 + \cos \theta }}
{{1 - \cos \theta }}[/tex]
 
  • #5
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okay, so iam here:
[tex]
\left( {\frac {a}{b+c}}+1 \right) \left( -{\frac {a}{b+c}}+1

\right) ^{-1}+ \left( 1+{\frac {b}{a+c}} \right) \left( 1-{\frac {b}

{a+c}} \right) ^{-1}+ \left( 1+{\frac {c}{a+b}} \right) \left( 1-{

\frac {c}{a+b}} \right) ^{-1}

[/tex]

is there any way that this can be equal to 0 !!

PS: sorry about messing up the equation in latex, but i am sure u still can get what i did (i am using maple 10 to convert to latex, i think there is a problem there)
 
  • #6
TD
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I calculated it too and I don't think it simplifies to 1, although I can't find an error in my derivation for the tangents to cosines.
 
  • #7
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Okay, and for the first one, I noticed that what I had is a sine rule... it's equal tu sin((b+c)/2) but, where do i get it to get equal to cos (a/2) ... :huh:
 
  • #8
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mohlam12 said:
Okay, and for the first one, I noticed that what I had is a sine rule... it's equal tu sin((b+c)/2) but, where do i get it to get equal to cos (a/2) ... :huh:
Very good, but we know that [itex]a+b+c=180[/itex] so:

[tex]\sin \left( {\frac{{a + b}}
{2}} \right) = \sin \left( {\frac{{180 - c}}
{2}} \right) = \sin \left( {90 - \frac{c}
{2}} \right) = \cos \left( { - \frac{c}
{2}} \right) = \cos \left( {\frac{c}
{2}} \right)[/tex]
 
  • #9
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yup, thanks!
 
  • #10
TD
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No problem :smile:

Never forget to use everythings that's given.
 
  • #11
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but anyone, can help me with the second problem?! pliiiiz
 
  • #12
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ok i got it. never mind everyone!
 

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