Problem 55 - Precalculus : Building Concepts and Connections

Question

Show that the Law of Cosines applied to a right triangle yields the Pythagorean Theorem.

Step-by-Step Solution

Answer

The Law of Cosines simplifies to the formulation of the Pythagorean Theorem when applied to a right triangle. This is because the cosine of a right angle (90 degrees) is zero, eliminating the third term in the law and reducing it to the formula described by the Pythagorean Theorem: \(c^2 = a^2 + b^2\) where \(c\) is the hypotenuse of the right triangle, and \(a\) and \(b\) are the other two sides.

Step 1: Recall the Law of Cosines

The Law of Cosines for a triangle with sides of lengths \(a\), \(b\), and \(c\), and an angle \(C\) opposite to side \(c\) is given by: \(c^2 = a^2 + b^2 - 2ab \cos C\).

Step 2: Apply the Law of Cosines to Right Triangle

In a right triangle, the angle \(C\) is 90 degrees. So, we can substitute this into the law of cosines. Note that the cosine of 90 degrees is 0, so the equation becomes: \(c^2 = a^2 + b^2 - 2ab \cos 90° = a^2 + b^2 - 2ab \cdot 0 = a^2 + b^2\).

Step 3: Compare with Pythagorean Theorem

The Pythagorean Theorem states \(c^2 = a^2 + b^2\), which is the same as the obtained formula, Therefore, it is showed that the Law of Cosines applied to a right triangle yields the Pythagorean Theorem.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry. It applies exclusively to right triangles, which are triangles with one angle measuring 90 degrees. This theorem establishes a relationship between the lengths of the sides of a right triangle. The formula is given by:

• \(c^2 = a^2 + b^2\)

Here, \(c\) represents the length of the hypotenuse (the side opposite the right angle), whereas \(a\) and \(b\) are the lengths of the other two sides. This relationship helps us calculate one side of a triangle if the lengths of the other two sides are known.
The Pythagorean Theorem is not just a special case but a crucial illustration of how trigonometry relates to spatial understanding in mathematics. Its implications are vast, being applied in diverse fields such as architecture, navigation, and physics.

right triangle

A right triangle is one of the simplest forms of a triangle, characterized by having one angle that is exactly 90 degrees. This property creates a unique set of rules and formulas applicable to right triangles.
In right triangles, the longest side, known as the hypotenuse, is opposite the right angle. The other two sides are often referred to as the 'legs' of the triangle.

• The Pythagorean Theorem specifically deals with right triangles.
• The hypotenuse is crucial for applying many trigonometric concepts such as sine, cosine, and tangent.

Understanding right triangles is vital because they often serve as building blocks for more complex geometrical shapes and problems. They also appear frequently in real-life situations, providing straightforward solutions with applications in engineering, construction, and various technical fields.

trigonometry

Trigonometry is a branch of mathematics that focuses on the relationships between the angles and lengths of triangles. It is especially useful in analyzing right triangles due to its various functions:

• Cosine (cos)
• Sine (sin)
• Tangent (tan)

In the case of a right triangle, the Law of Cosines simplifies greatly, directly leading to the familiar Pythagorean Theorem. This happens because the cosine of a 90-degree angle is zero. Thus, in a right triangle, the Law of Cosines simplifies to:\
\(c^2 = a^2 + b^2 - 2ab \cdot \cos 90° = a^2 + b^2\)
Trigonometry not only links back to simpler geometric principles like the Pythagorean Theorem but also extends to a vast range of applications, from computing distances in physics to modeling periodic phenomena in engineering and nature. It creates a mathematically beautiful and computationally powerful framework for dealing with angles and lengths.