# Finding Value $q$ in a Triangle Given: Base $c$, Height $hc$, and Angle $γ$

The objective is to determine the value $$q$$ for a specific angle $$γ$$.

Given is a triangle with the known values:

• base length $$c$$,
• height $$hc$$ relative to this base,
• the angle $$γ$$ opposite this base.

The challenge arises due to the ambiguity created by the triangle inequality and the fact that, for a given value of $$γ$$, there are two possible configurations of the triangle. This results in two potential values for $$q$$: $$q$$ and $$-q$$

Approach So Far: Various trigonometric relationships and triangle properties have been utilized to express $$q$$, but it has not been possible to derive a closed form for $$q$$ solely in terms of $$c$$, $$hc$$, and $$γ$$. With more given Values it is possible to calculate $$q$$ with a specific angle $$γ$$:

$$q = \tan(90°-α) ⋅ a ⋅ \sin(α-γ)$$

but this way, I can only calculate the given triangle with angle $$α$$ and $$a$$

The two possible configurations:

Let's start with the area: $$A=\frac{1}{2}c \cdot h=\frac{1}{2}ab \cdot \sin \gamma \implies b=\frac{ch}{a \sin \gamma}$$
Next, apply law of cosine: $$c^2=a^2+b^2-2ab\cos \gamma$$, substituting $$b$$ we get the equation to find $$a^2$$: $$c^2=a^2 + \frac{c^2h^2}{a^2\sin ^2 \gamma}-2ch \cot \gamma$$
This is a quadratic equation in terms of $$a^2$$, so we can express $$a^2:$$ $$a^2=0.5 \cdot \left(2ch\cot \gamma+c^2+\sqrt{(2ch\cot \gamma+c^2)^2-\frac{4c^2h^2}{\sin ^2 \gamma}}\right)$$ Note that the problem is symmetrical relative to $$a$$ and $$b$$ so the solution will cover both possible configurations. Also, if the expression under the square root is negative, there is no solution which takes care of the triangle inequality.
Finally, $$q^2=a^2-h^2$$