The Law of Cosines can be derived by applying some basic algebra. In a triangle, for example, the obtuse angle of a side is larger than that of the opposite side. It follows that the obtuse angle’s square is twice as large as its perpendicular. In other words, the obtuse angle of a side is twice as large as its perpendicular, and the same applies to an acute angle.
The Law of Cosines is derived in two ways. The first way is to apply it to the Pythagorean Theorem. If you substitute ‘b’ for ‘a’, you get the same result as before. Now, you can apply the Law of Cosines to calculate missing angles. When C is a right angle, the Law of Cosines is identical to the Pythagorean Theorem.
The second way to apply the Law of Cosines is to solve a problem. For example, a triangle with side a is a right triangle. In order to derive the obtuse angle, replace a by b, and c by a. Then you’ll have the cosine angle and the right angle. The Law of Cosines is useful in navigation, surveying, astronomy, and geometry.
A final way to prove the Law of Cosines is to calculate the area of a parallelogram. A parallelogram with a negative angle is obtuse. The area of the obtuse parallelogram is equal to the area of the ABC triangle. The same goes for auxiliary triangles. If you have an auxiliary triangle with an obtuse angle, the area of that triangle is equal to the area of a parallelogram with the acute angle.
