Sources: http://www.mathwarehouse.com/geometry/triangles/area/herons-formula-triangle-area.php
https://www.mathsisfun.com/algebra/trig-cosine-law.html
Using the law of Cosines:
\(C^2 = A^2 +B^2 - 2ABcos(c)\)
Where C is the unknown side
A is one the known sides
B is the other known side
c is the angle between the two known sides
Now plug in values to find C
\(C^2 = 5^2 + 8^2 - 2(5)(8)cos105\) = \(109.7055 \)
\(\sqrt(C^2)= C\)
\(\sqrt(109.7055)\) = \(10.474 = 10.5 = C \)
Finding the area (using Heron's Formula);
\(Area= \sqrt(S(S-A)(S-B)(S-C)\)
where S is the semi-perimeter (half the perimeter)
A is one of the sides
B is another side
C is the final side
Now to plug it in:
The area is 19.9