Problem 1 - Precalculus

Question

Carry out the indicated operations. (a) \(-S C+12 S C\) (b) \(-\sin \theta \cos \theta+12 \sin \theta \cos \theta\)

Step-by-Step Solution

Answer

(a) 11SC; (b) 11\sin \theta \cos \theta.

Step 1: Observe the Expression

For both expressions (a) and (b), start by observing that they are linear expressions with like terms. This means we can combine the terms by simple addition or subtraction.

Step 2: Combine Like Terms for (a)

In expression (a), combine the like terms \(-S C\) and \(+12 S C\).\ The coefficient of \(SC\) in \(-SC\) is \(-1\), so you perform the operation \(-1 + 12\), which equals \(+11\). Therefore, the expression becomes \(+11 S C\).

Step 3: Combine Like Terms for (b)

In expression (b), the like terms \(-\sin \theta \cos \theta\) and \(+12 \sin \theta \cos \theta\) can be combined.\ The coefficient for \(\sin \theta \cos \theta\) is \(-1\) in \(-\sin \theta \cos \theta\), so the operation \(-1 + 12\) equals \(+11\). Therefore, the expression simplifies to \(+11 \sin \theta \cos \theta\).

Key Concepts

Understanding Like Terms

In algebra, like terms are terms that have identical variable parts. This means their variable components, including both the variable itself and any exponents, are the same. For example, in the expression

• \(-S C + 12 S C\), both terms are considered like terms because they share the same variable part, \( S C \).
• Similarly, in expression \(-\sin \theta \cos \theta + 12 \sin \theta \cos \theta\), the term \( \sin \theta \cos \theta \) remains identical, making both parts like terms.

To combine like terms, you simply need to add or subtract their coefficients while maintaining the common variable part. This is a fundamental feature that simplifies the process of dealing with expressions, allowing you to combine terms into a simpler expression with ease.

Simplifying Expressions

Simplifying an expression involves reducing it to its most straightforward form, which can often involve combining like terms. This makes working with and understanding expressions easier. When you have identified like terms, as described earlier,

• you can focus on the numerical coefficients of these terms.
• For example, in the case of the linear expression \(-S C + 12 S C\),
• you combine the coefficients of these like terms: \(-1 + 12 = 11\).

The resulting expression simplifies to \(11 S C\). For trigonometric expressions, such as \(-\sin \theta \cos \theta + 12 \sin \theta \cos \theta\), the process is much the same.

• The coefficients, \(-1 + 12\),
• give you an expression of \(11 \sin \theta \cos \theta\).

The goal of simplification is to make the expression easier to use or evaluate and to help see the relationships between different variables or components more clearly.

Introduction to Linear Expressions

Linear expressions are algebraic expressions where each term is either a constant or a product of a constant and a single variable raised to the power of one. This makes them straightforward to work with because they do not involve products of variables or variables raised to higher powers. In the examples given:

• \(-S C + 12 S C\)
• and \(-\sin \theta \cos \theta + 12 \sin \theta \cos \theta\),

each one collectively forms a linear expression. The absence of variables being squared or multiplied by each other simplifies calculations and transformations. By their nature, linear expressions can be easily manipulated through addition and subtraction, making it easier to find like terms and simplify. They form fundamental building blocks in algebraic manipulations and are a key concept for more complex equations and trigonometric identities.