Problem 102 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The double-angle identities are derived from the sum identities by adding an angle to itself.

Step-by-Step Solution

Answer

The statement makes sense due to the following reasoning: In trigonometry, double angle identities can indeed be derived from the sum identities by adding an angle to itself. This is because \(2x = x + x\).

Step 1: Understanding the Trigonometric Identities

First, we need to understand what double-angle and sum identities represent in trigonometry. Double angle identities are associated with functions of double angles, i.e., \(2x\). On the other hand, sum identities represent the functions of the sum of two angles, i.e, \(x+y\).

Step 2: Comparison between Double-angle and Sum Identities

Having understood the concepts of double-angle and sum identities, now let's relate them to the given statement - \'The double-angle identities are derived from the sum identities by adding an angle to itself\'. This statement can be interpreted as follows: for any angle \(x\), the double angle \(2x\) is indeed the sum of the angle added to itself, i.e., \(x + x\). This means that it's possible to arrive at a double angle identity from a sum identity.

Step 3: Final Conclusion on the Statement

Given the insights from Steps 1 and 2, it can be concluded that the statement provided makes sense. It is indeed possible to obtain the double-angle identities from the sum identities by adding an angle to itself because \(2x = x + x\).