Proving Triangle Properties with Sin and Cos

In summary, In this conversation, the author explains that there is a relationship between the angles in a triangle and the cosine of the angle. For the first problem, the author does not know how to solve it, and for the second problem, the author calculates that tan²(α/2) + tan²(β/2) + tan²(γ/2) = 1.
  • #1
mohlam12
154
0
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 don't 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 don't know what to do! if someone can help me out! or maybe give me hints...

i appreciate your help!
 
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  • #2
mohlam12 said:
this one, i erally don't 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
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 going to use LaTex, I can't get what I want...

tan²(α/2) + tan²(β/2) + tan²(γ/2) = 1

Thanks,
 
  • #4
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 going to 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
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
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
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
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
yup, thanks!
 
  • #10
No problem :smile:

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

1. What are the basic properties of a triangle?

The basic properties of a triangle include having three sides, three angles, and a total sum of 180 degrees. Additionally, the longest side of a triangle is always opposite the largest angle, and the shortest side is always opposite the smallest angle.

2. How can I prove triangle properties using sine and cosine?

Sine and cosine are trigonometric functions that can be used to calculate the lengths of sides and measures of angles in a triangle. By using the properties of these functions, such as the Pythagorean identity, you can prove various properties of triangles.

3. What is the Law of Sines and how is it used to prove triangle properties?

The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all sides and angles in the triangle. This law can be used to prove properties such as the relationship between the sides and angles of similar triangles.

4. Can I use cosine to prove triangle properties?

Yes, cosine can also be used to prove triangle properties. For example, the Law of Cosines can be used to find the length of a side or the measure of an angle in a triangle when the other sides and angles are known.

5. What are some common triangle properties that can be proved using sine and cosine?

Some common triangle properties that can be proved using sine and cosine include the Pythagorean Theorem, the Law of Sines and Cosines, the relationships between the sides and angles of similar triangles, and the properties of right triangles such as the trigonometric ratios (sin, cos, and tan).

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