Solving a triangle by angle, circumradius, and area

Question

The problem itself is as follows.

Consider a triangle $ABC$. The cosine of one of its interior angles is $m$, its circumradius is equal to $R$, and its area is equal to $S$. Solve this triangle: find its sides and angles.

By applying the sine theorem, I've got that the length of a side, opposite to the angle with the known cosine, which is $2R\sqrt{1-m^2}$. But I had no luck in advancing further, as it either results in too many unknowns. Most I could do is getting the product of sines of two unknown angles, from the formula $S=2R^2\sin\angle A\sin\angle B\sin\angle C$.

I'm looking for proofs that such triangle exists and is clearly defined, as well as the way to solve it if it is.

Answer 1

This is a sketch of a solution. Let's say wlog that $\cos \angle A=m$. We have $a=2R\sqrt{1-m^2}$. We also have $2A=bc \sqrt{1-m^2}$ and $b=2R\sin \angle B$, $c=2R\sin \angle C$. On the other hand, $a^2=c^2+b^2-2bc \cdot m$.

Thus, $4R^2(1-m^2)=4R^2\sin^2 \angle B+4R^2\sin^2 \angle C-\frac{4A\cdot m}{\sqrt{1-m^2}}$.
Finally, $\sin \angle C=\sin (\angle A+\angle B)=m\sin \angle B+\sqrt{1-m^2}\cos \angle B$. Substituting this into above will give us equation to find $\sin \angle B$.

Answer 2

Below is my own answer on the problem I came up with. There may exist more elegant ways, but this is the most straightforward way of which I know.

Consider a triangle $ABC$ with circumradius $R$, area $S$, and $\cos\angle C=m$.

1. Find the sine of $\angle C$ by using the famous trigonometric identity $\sin^2\alpha+\cos^2\alpha=1$. As all interior angles of the triangle have positive sines, we take the positive value, thus $\sin\angle C=\sqrt{1-\cos^2\angle C}=\sqrt{1-m^2}$.
2. Find the length of side $AB$, opposite to $\angle C$, by using the law of sines. Thus, $AB=2R\sin\angle C=2R\sqrt{1-m^2}$.
3. Find the product of two other sides. For that, find the altitude of triangle $ABC$ to side $AB$ by using the formula $h_{AB}=\dfrac{2S}{AB}$. Now, if we equate two formulas for the area of a triangle, we get $\dfrac{AB\cdot BC\cdot AC}{4R}=\dfrac{1}{2}AB\cdot h_{AB}$, which we can transform further: $$\begin{align*} \dfrac{AB\cdot BC\cdot AC}{4R} = \dfrac{1}{2}AB\cdot h_{AB}\ \ &\Leftrightarrow\ \ \dfrac{BC\cdot AC}{2R} = h_{AB} \\ &\Leftrightarrow\ \ \dfrac{BC\cdot AC}{2R} = \dfrac{2S}{AB} \\ &\Leftrightarrow\ \ BC\cdot AC = \dfrac{4RS}{AB} \end{align*}$$ Thus, $BC = \dfrac{4RS}{AB\cdot AC}$.
4. By using the law of cosines, we get $AB^2=BC^2+AC^2-2\cdot BC\cdot AC\cos\angle C$. Here, we can substitute $BC\cdot AC = \dfrac{4RS}{AB}$, and $BC = \dfrac{4RS}{AB\cdot AC}$. We get: $$\begin{align*} AB^2=\left(\dfrac{4RS}{AB\cdot AC}\right)^2+AC^2-2\cdot \dfrac{4mRS}{AB} \end{align*}$$

Here, we know everything except $AC$, and we can find both unknown sides by simplifying the last formula into the biquadratic equation, positive roots of which are sides $BC$ and $AC$. Finally, to find the other angles, one can use the law of cosines if the given angle is obtuse ($m < 0$), as there is only one obtuse angle in the triangle, or the law of cosines in all other cases.