Problem 112 - College Algebra

Question

What is an identity? Give an example.

Step-by-Step Solution

Answer

An identity is a mathematical equation or inequality that is always true, irrespective of the variables' values. An example is the trigonometric identity \(\sin^2(x) + \cos^2(x) = 1\), which is valid for all real numbers \(x\).

Step 1: Define an Identity

An identity in mathematics is an equation or inequality that is always true, regardless of the values of its variables. In other words, substituting any legitimate value for the variables in an identity will produce a true statement.

Step 2: Give an Example of an Identity

The elementary trigonometric identity \(\sin^2(x) + \cos^2(x) = 1\) is a good example. This equation is true for all real numbers \(x\). You can substitute any number for \(x\), and the left-hand side of the equation will always equal the right-hand side.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within the functions' domains. These identities are fundamental tools in trigonometry, allowing us to simplify complex expressions, solve trigonometric equations, and compute angles in geometric problems.

For instance, one of the most well-known trigonometric identities is the Pythagorean identity: \[\sin^2(x) + \cos^2(x) = 1\]. No matter what value you substitute for \(x\), this relationship remains constant, underpinning the inherent properties of sine and cosine functions in relation to the right triangle. Understanding and applying these identities can vastly improve a student's ability to work with trigonometric equations and deepen their comprehension of trigonometric concepts.

Algebraic Equations

Algebraic equations are statements of equality that contain one or more variables. A solution to an algebraic equation is a value or set of values that, when substituted for the variable, make the equation a true statement. These equations are the building blocks of algebra and form the basis for more complex mathematical concepts.

Simple algebraic equations can often be solved by isolating the variable on one side of the equation. For example, in the equation \(2x + 3 = 7\), you would subtract 3 from both sides and then divide by 2 to find that \(x = 2\). This process is just the beginning; more complex equations might require factoring, using the quadratic formula, or working with higher-degree polynomials. Mastery of algebraic equations enables students to progress to more advanced mathematics with confidence.

Equations and Inequalities

Equations and inequalities are two fundamental concepts that represent relationships between expressions. An equation indicates that two expressions are equivalent, while an inequality shows that one expression is greater than or less than another.

When solving inequalities, it is crucial to remember that multiplying or dividing both sides by a negative number reverses the inequality sign. For example, if we have \(-2x > 4\), dividing both sides by -2 and reversing the inequality gives us \(x < -2\). Inequalities are not only central in algebra but also play a critical role in calculus, optimization and other fields of mathematics. Students learning to solve inequalities gain insights into the behavior of mathematical relationships and their practical applications in areas such as economics and engineering.