Can You Find All Triangle Angles from 2 Sides?

[skrat]

Homework Statement

Is it possible to determine all the angles in a triangle, if we only know the length of two sides?

Homework Equations

The Attempt at a Solution

I was thinking for quite some time and I don't think it is possible. It probably is, if two sides are peprendicular but if not, I don't think so.

 

[PhysicoRaj]

Think about how you relate the sides and angles of a triangle?

 

[skrat]

You mean the definition of the dot product?

$\vec{a}\cdot \vec{b}=\left \| \vec{a} \right \|\left \| \vec{b} \right \|cos\theta$

Would be great yeah, but I don't have the coordinates. I only have the length of the sides.

 

[PhysicoRaj]

A modified version of that dot product called the cosine rule comes in handy for this. Have you studied this?

 

[skrat]

I have.

$c^2=a^2+b^2-2abcos\theta$

But this is a "system" of one equations with two parameters. How would you reduce the number of parameters, or better; how would you find the length of the third side?

 

[PhysicoRaj]

Don't you know all the sides, as in the problem?

 

[skrat]

Hah. Ok, there is a mistake in the original post. I apologize.
I only know the length of TWO sides. (I will edit my first post)

 

[PhysicoRaj]

You can use the law of Sines as well. I think you can eliminate the third side by using an expression for it derived from law of sines.

 

[skrat]

I can eliminate the third side but than I get another angle inside the equation.

$\frac{a}{sin\alpha }=\frac{b}{sin\beta }=\frac{c}{sin\gamma }$
and
$c^2=a^2+b^2-2abcos\gamma$

gives me $(a\frac{sin\gamma }{sin\alpha })^2=a^2+b^2-2abcos\gamma$

 

[Quesadilla]

Let $b, c$ be two sides of a triangle with known lengths and let $\alpha$ be the angle between them. Now consider each $\alpha \in (0, \pi)$.

 

[muthukrishnan]

Not at all possile to know the angle of triangle with two sides known.There will e infinite number of solutions .
Just think how will you first draw the trianle with two lengths are known.First draw one line whose length is known.Then try to draw the second line starting from on edge of the first line.This second line can be drawn at any angle zero to 360 deg.So that will result in infinite number of lines .So finally finished triange will have will have infinite solutions.

 

[Char. Limit]

If you know the length of two sides and the angle between those two, you can figure it out. If you know the length of two sides and the angle between one of them and the third side, you can narrow it down to two possibilities. If you don't know /any/ angles, though, there's nothing you can do.

 

[skrat]

Yup, I thought this may be the case yeah. :/

Ok, thanks!