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Chapter 6: Problem 60

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

Short Answer

Expert verified
The Law of Cosines becomes the Pythagorean Theorem when the angle \(C\) in the law of cosines is \(90^\circ\), making \(\cos(C) = 0\). So, \(c^2 = a^2 + b^2 - 2ab \times 0\) simplifies to \(a^2 + b^2 = c^2\).

Step by step solution

01

Introduction of the Pythagorean theorem

The Pythagorean theorem is a fundamental relation in Euclidean geometry. It states: In a right triangle, the square of the length of the hypotenuse \(c\) (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides \(a\) and \(b\). It's represented by the formula: \(a^2 + b^2 = c^2\) .
02

Introduction of the Law of Cosines

The Law of Cosines is used to calculate an unknown side or an unknown angle in a triangle, it states: The square of the length of a 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 these two lengths and the cosine of the included angle. It's represented by the formula: \(c^2 = a^2 + b^2 - 2ab \cos(C)\), where C is the angle included between sides a and b.
03

Show the Relationship

The relationship between these two is that when the triangle in the law of cosines is a right triangle, angle \(C\) will be \(90^\circ\). The cosine of \(90^\circ\) is 0. So, the law of cosines simplifies to: \(c^2 = a^2 + b^2 - 2ab \times 0\), which simplifies to \(a^2 + b^2 = c^2\). Hence, in the case of right triangles, the Law of Cosines reduces to the Pythagorean Theorem.

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