To solve the for the angles when we have the lengths of all three sides, we use the law of cosines.
The law of cosines states that angle A = cos^-1((a^2 - b^2 - c^2)/(-2bc)), B = cos^-1((b^2 - a^2 - c^2)/(-2ac)), and C = cos^-1((c^2 - a^2 - b^2)/(-2ab)).
Let's find angle A first:
A = (a^2 - b^2 - c^2)/(-2bc)
A = (12^2 - 15^2 - 20^2)/(-2(15)(20))
A = cos^-1((-481)/(-600))
A ~~ 36.71^@
Now angle B:
B = cos^-1((b^2 - a^2 - c^2)/(-2ac))
B = cos^-1((15^2 - 12^2 - 20^2)/(-2(12)(20)))
B = cos^-1((-319)/(-480))
B ~~ 48.34^@
Finally angle C:
C = cos^-1((c^2 - a^2 - b^2)/(-2ab))
C = cos^-1((20^2 - 12^2 - 15^2)/(-2(12)(15)))
C = cos^-1((31)/(-360))
C ~~ 94.93^@
We can also find angle C by doing 180^@ - 36.71^@ - 48.34^@, since the measures of the angles in a triangle add up to 180^@.
Hope this helps!
