find the "X"

JeffM said:
How is the x that you are supposed to find defined?

What have you tried?
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Is it possible that x\displaystyle x is the side of the triangle?
Is it possible you know the law of cosines?

Show us your work! WE don't know what YOU know!
 
DrPhil said:
Is it possible that x\displaystyle x is the side of the triangle?
Is it possible you know the law of cosines?

Show us your work! WE don't know what YOU know!
Click to expand...
yes i know the law of cosines.
the X is the side of triangle.
 
slice_of_god said:
yes i know the law of cosines.
the X is the side of triangle.
Click to expand...
04bupjr5.jpg
There are three triangles in which x2\displaystyle x^2 can be found as functions of angles α, β, γ\displaystyle \alpha,\ \beta,\ \gamma, using the Law of Cosines.
A fourth equation is
........α+β+γ=2π\displaystyle \alpha+ \beta +\gamma = 2\pi
In principle, that is four (independent) equations in four unknowns, so a solution exists.

How to find the solution requires either trig, geometric construction, or lucky guesswork. Part of the trickiness may be how to construct a 3-4-5 triangle, in which you would recognize the angle between sides of 3 and 4 is is a right angle.

I started looking at it from the trig point of view.

Note that α=2π−(β+γ)\displaystyle \alpha = 2\pi - (\beta+\gamma), from which
.........cos⁡α=cos⁡(β+γ)=cos⁡βcos⁡γ−sin⁡βsin⁡γ\displaystyle \cos\alpha = \cos(\beta + \gamma) = \cos\beta\cos\gamma - \sin\beta\sin\gamma
That leads to some complicated algebra, but might lead to an equation for x\displaystyle x in closed form.
It is also true that
.........sin⁡α=−sin⁡(β+γ)=−(sin⁡βcos⁡γ+cos⁡βsin⁡γ)\displaystyle \sin\alpha = -\sin(\beta + \gamma) = -(\sin\beta\cos\gamma + \cos\beta\sin\gamma)
[I don't know if that is any help or not.]

Did you look up the geometric solution that Denis found?
Denis said:
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A hint from that solution is that α=90°+60°=150°\displaystyle \alpha = 90° + 60° = 150°. If you can prove that somehow, then the problem is solved.

These are only some musings - what have YOU tried?
 
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