Problem 87 - Precalculus Essentials

Question

The distance formula and the definitions for cosine and sine are used to prove the formula for the cosine of the difference of two angles. This formula logically leads the way to the other sum and difference identities. Using this development of ideas and formulas, describe a characteristic of mathematical logic.

Step-by-Step Solution

Answer

The characteristic of mathematical logic that can be seen from the derivation of trigonometric identities is coherence and structure. A single identity, such as the formula for the cosine of the difference of two angles, can serve as a starting point for establishing other identities. Mathematical theories or rules are many times built from simpler, previously established ones.

Step 1: Understand the basic formula

The cosine of the difference of two angles formula is defined as \(cos(a - b) = cos(a)cos(b) + sin(a)sin(b)\), a foundational formula in trigonometry. This formula provides a way to express the cosine function of the difference of two angles in terms of cosine and sine functions of those angles.

Step 2: Derive other identities

The formula for cosine of the sum of two angles, \(cos(a + b) = cos(a)cos(b)−sin(a)sin(b)\), can be derived by letting \(b = -b\) in the cosine of the difference formula. Likewise, the formulae for sine of the sum of two angles (\(sin(a + b) = sin(a)cos(b) + cos(a)sin(b)\)) and sine of the difference of two angles (\(sin(a - b) = sin(a)cos(b) - cos(a)sin(b)\)) can be derived.

Step 3: Describe characteristic of mathematical logic

The characteristic of mathematical logic that is observed here is coherence and structure. Once a fundamental rule or identity is established, it can be used to derive additional rules. This principle of coherent structure and rules built upon previous rules is central to mathematical logic.