We will prove the Law of Sines for the acute triangle case and the obtuse triangle case.
If we want to prove the Law of Sines holds true in this acute case, it is enough to show that.
Now we have proved the Law of Sines for the Acute Triangle case!
continue and try to prove the Law of Sines for the Obtuse Triangle
Let be an obtuse triangle with . Notice again that h is the perpendicular from the point B to the line segment AC. H intersects the segment AC at the point D.
Alpha is an obtuse angle and Beta is its supplement. Remember that angles that are supplementary form a line. Therefore, alpha plus Beta form the line with points A,C, and D on it.
and so h = bsin and h=asin.
Setting the two h's equal to one another, we get
Therefore, and the Law of Sines works for the Obtuse Triangle case as well:)
We will prove the Law of Cosines for the acute triangle case and the obtuse triangle case.
Remember the cosine of an angle is equal to the .
Segment CD or h is the perpendicular from the point C to the line segment AB. It intersects the line segment AB at the point D.
. Therefore, .
we have proved the Law of Cosines for the Acute Triangle case.
Using the fact that Triangle ADB and Triangle CDB are both right triangles, we can use the Pythagorean Theorem to get...
Then finally, . So, we have proved the Law of Cosines for the Obtuse Triangle Case.