What familiar formula can you obtain when you use the third form of the Law of Cosines c^2 = a^2 + b^2 - 2ab cos C and you let C = 90deg? What is the relationship between the Law of Cosines and this formula?
cos C=0 and you just have the Pythagorean Theorem.
When we let C = 90 degrees in the third form of the Law of Cosines, c^2 = a^2 + b^2 - 2ab cos C, we obtain a familiar formula called the Pythagorean theorem.
Let's see how we can get this result:
First, substitute C = 90 degrees into the equation:
c^2 = a^2 + b^2 - 2ab cos 90
Since the cosine of 90 degrees is 0, the equation simplifies to:
c^2 = a^2 + b^2 - 2ab(0)
Simplifying further:
c^2 = a^2 + b^2
This equation is precisely the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).
So, we can say that the Pythagorean theorem is a special case of the Law of Cosines when the angle opposite to the side you are solving for is 90 degrees. The Law of Cosines provides a more general formula for finding the lengths of sides in any triangle, including those that are not right-angled.