Problem 70 - Precalculus with Limits

Question

THINK ABOUT IT What familiar formula do 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 \(C = 90^{\circ}\)? What is the relationship between the Law of Cosines and this formula?

Step-by-Step Solution

Answer

The Law of Cosines simplifies to the Pythagorean theorem when the angle is 90 degrees. Hence, the Law of Cosines can be seen as a generalization of the Pythagorean theorem.

Step 1: Substitute the angle value

In the equation \(c^2 = a^2 + b^2 - 2ab\ \cos\ C\), substitute \(C = 90^{\circ}\). The equation becomes \(c^2 = a^2 + b^2 - 2ab\ \cos\ 90^{\circ}\).

Step 2: Apply cosine of 90 degrees

As the cosine of 90 degrees is 0, the equation becomes \(c^2 = a^2 + b^2 - 2ab * 0\). After multiplication, it becomes \(c^2 = a^2 + b^2 - 0\).

Step 3: Simplify the expression

Simplifying the expression gives: \(c^2 = a^2 + b^2\). This is the Pythagorean theorem.

Step 4: Identify the relationship

The Law of Cosines, when the angle is 90 degrees, simplifies to the Pythagorean theorem. Hence, we can say the Law of Cosines is a generalized form of the Pythagorean theorem.

Key Concepts

Pythagorean theorem

The Pythagorean theorem is one of the most well-known principles in geometry. It relates the three sides of a right triangle. According to this theorem, in a right triangle with sides labeled as \(a\), \(b\), and \(c\), where \(c\) is the hypotenuse (the side opposite the right angle), the relationship is expressed as: \[c^2 = a^2 + b^2.\]This means that the square of the hypotenuse is equal to the sum of the squares of the other two sides. This fundamental theorem is not only essential for solving many geometric problems but also has applications in various disciplines, like physics and engineering.

• Works only in right-angled triangles.
• Basis for many proofs and derivations in geometry.
• Forms the foundation for the concept of distance in coordinate geometry.

The Pythagorean theorem connects directly to the Law of Cosines, particularly when the angle \(C\) is a right angle (90 degrees). When you set \(C = 90^{\circ}\) in the Law of Cosines, it simplifies to the Pythagorean theorem.

trigonometry

Trigonometry deals with the relationships between the angles and sides of triangles. Central to trigonometry are the trigonometric functions such as sine, cosine, and tangent. These functions help to understand angular relationships and are pivotal in solving various triangle-related problems.The Law of Cosines is essentially a trigonometric identity that extends beyond the right triangle. It states that for any triangle with sides \(a\), \(b\), and \(c\), and angle \(C\), the relationship is:\[c^2 = a^2 + b^2 - 2ab \cos C.\]This formula allows for calculations of unknown sides or angles and reduces to the Pythagorean theorem when \(C\) is \(90^{\circ}\) because the \(\cos 90^{\circ} = 0\). In other words, trigonometric rules like the Law of Cosines demonstrate the flexibility of trigonometric principles in handling both right and non-right triangles.

• Encapsulates the functions sine, cosine, and tangent.
• Helps predict triangle side lengths and angles.
• Enables the transition from the Law of Cosines to geometrical interpretations.

geometry

Geometry is a vast field of mathematics that deals with shapes, sizes, and the properties of space. Within geometry, triangles play a critical role, and understanding their properties is essential for broader geometric comprehension. The Law of Cosines bridges trigonometry and geometry by providing a tool that helps analyze the properties of triangles. By understanding how the sides are related through an angle, it introduces a comprehensive way to look at triangle geometry. When applied, the Law of Cosines is immensely useful in:

• Calculating remaining sides in non-right triangles.
• Determining angles when all sides are known.
• Solving complex geometric shapes by breaking them into triangles.

By allowing calculations in non-right triangles, the Law of Cosines extends geometric understanding beyond the scope of the Pythagorean theorem. It shows how mathematical rules can provide deeper insight into more complex shapes and configurations in space.