Problem 60 - Algebra and Trigonometry

Question

Explain why the Pythagorean Theorem is a special case of the Law of Cosines.

Step-by-Step Solution

Answer

The Pythagorean theorem is a special case of the Law of Cosines because when the included angle is 90 degrees (which is the case in a right triangle), the Law of Cosines simplifies to the Pythagorean theorem.

Step 1: Define the Pythagorean theorem

The Pythagorean theorem states that in any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. If a, b are the two sides and c is the hypotenuse, we write it as \( a^2 + b^2 = c^2 \).

Step 2: Define the Law of Cosines

The Law of Cosines generalizes the Pythagorean theorem, working with any triangle, not only right triangles. It states that the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, minus twice the product of the two other sides multiplied by the cosine of the included angle. If a, b, c are the sides and \(\gamma\) is the angle included by a and b, it's written as \( c^2 = a^2 + b^2 - 2ab \cos(\gamma) \).

Step 3: Explain why the Pythagorean theorem is a special case of the Law of Cosines

In the case of a right triangle, the included angle \(\gamma\) between sides a and b is 90 degrees. Cosine of 90 degrees is zero. So when we plug \(\gamma = 90\) into the Law of Cosines, the formula simplifies to \( c^2 = a^2 + b^2 - 2ab \cos(90) = a^2 + b^2 \), which is exactly the Pythagorean theorem. So we can see the Pythagorean theorem is a special case of the Law of Cosines, applying when we have a right triangle.