Calculate All Sides and Angles from Area, One Side, and Adjacent angle of triangle.

Question

I have one Area of Triangle, One adjacent angle, and one side. Is it possible to calculate all other sides and angles ?

I have searched this problem, but could not get any reference to it.

Answer 1

Note that if $a$ and $b$ are two sides of the triangle and $\gamma$ is the angle in between then its Area is $$\text{Area}=\frac{a\cdot b\cdot\sin(\gamma)}{2}\implies b=\frac{2\cdot \text{Area}}{a\cdot \sin(\gamma)}$$ and we find another side. For the third side use the Heron's formula or the Law of Cosines and finally find the remaining angles from the Law of Sines. Can you take it from here?

Answer 2

Suppose in $\triangle ABC$ that the known angle is $A$ and the adjacent known side is $b$. We can get the altitude $h_c$ from vertex $C$ onto $AB$ as $$h_c=b\sin A$$ The length of $AB$, or $c$, can now be determined as $\frac{2\operatorname{Area}(ABC)}{h_c}$. Now two sides and the included angle are known, so the rest is easy.