Geometrical solution to static problem without friction

[maros522]

Homework Statement

We have triangle with sides d1,d2,l and angle \alpha between d1 and d2.
Assume small change \Delta\alpha of \alpha.

Homework Equations

Then we can write for \Deltal equation
\Deltal=(d1*d2)/(l) * sin\alpha\Delta\alpha.

How can I prove that?

The Attempt at a Solution

I start with function cos(\alpha+\Delta\alpha)=cos\alpha * cos\Delta\alpha - sin\alpha * sin\Delta\alpha
and laws of cossines (l+\Delta l)^2=d1^2+d2^2-2d1d2*cos(\alpha+\Delta\alpha)
But I don't know how to continue. :(

 

[rock.freak667]

The Attempt at a Solution

I start with function cos(\alpha+\Delta\alpha)=cos\alpha * cos\Delta\alpha - sin\alpha * sin\Delta\alpha
and laws of cossines (l+\Delta l)^2=d1^2+d2^2-2d1d2*cos(\alpha+\Delta\alpha)
But I don't know how to continue. :(

So if Δα is small, then sin(Δα) = Δα and cos(Δα) =1.

So you can simplify cos(Δα+α) using that fact.

Now if you use the original triangle with sides d₁,d₂,l and angle α, what is l² using the cosine rule?

(l+\Delta l)^2=d1^2+d2^2-2d1d2*cos(\alpha+\Delta\alpha

If you expand the left side and substitute for cos(α+Δα) and you know that Δl is small so (Δl)² is negligible you should get it easily.

 

[maros522]

Thank you for answer I get it now.