Problem 88 - Precalculus Essentials

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The word identity is used in different ways in additive identity, multiplicative identity, and trigonometric identity.

Step-by-Step Solution

Answer

All the statements make sense: In the additive context, the identity is the number that doesn't change other numbers when added (0). In the multiplicative context, it's the number that doesn't change other numbers when multiplied (1). In the trigonometric context, it's a mathematical equation that holds true for any values of the variables. The word 'identity' is used correctly in all three contexts.

Step 1: Understanding Additive Identity

In mathematics, an additive identity is an element that, when added to any number, doesn't change the value of the number. For example, zero is the additive identity because adding zero to any number doesn't alter the number's value. The statement makes sense.

Step 2: Understanding Multiplicative Identity

A multiplicative identity is an element that when multiplied with any number, doesn't change that number's value. The number 1 serves as the multiplicative identity because any number times 1 remains unchanged. The statement about multiplicative identity also makes sense.

Step 3: Understanding Trigonometric Identity

A trigonometric identity is a mathematical equation that is always true for any values of the variables where both sides of the equation are defined. Some common examples are \( \sin^2(x) + \cos^2(x) = 1 \) or \( 2 \sin(x) \cos(x) = \sin(2x) \). Thus, the statement regarding trigonometric identity is sensible.