Problem 2 - Precalculus

Question

Carry out the indicated operations. (a) \(10 S C+4 S C-16 S C\) (b) \(10 \sin \theta \cos \theta+4 \sin \theta \cos \theta-16 \sin \theta \cos \theta\)

Step-by-Step Solution

Answer

(a) -2SC, (b) -2 \sin \theta \cos \theta.

Step 1: Identify Like Terms in Expression (a)

The expression given is \(10SC + 4SC - 16SC\). All terms contain the same variable combination \(SC\).

Step 2: Combine Like Terms for Expression (a)

Add the coefficients of the like terms together: \(10 + 4 - 16\). The result is \(-2\). Therefore, the simplified expression is \(-2SC\).

Step 3: Identify Like Terms in Expression (b)

The expression is \(10 \sin \theta \cos \theta + 4 \sin \theta \cos \theta - 16 \sin \theta \cos \theta\). Each term contains the same trigonometric function combination \(\sin \theta \cos \theta\).

Step 4: Combine Like Terms for Expression (b)

Add the coefficients of the like terms: \(10 + 4 - 16 = -2\). The simplified expression is \(-2 \sin \theta \cos \theta\).

Key Concepts

Trigonometric Expressions

Trigonometric expressions involve functions like sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), which are fundamental in studies of triangles and waveforms. In our given exercise, the trigonometric expression is \( 10 \sin \theta \cos \theta + 4 \sin \theta \cos \theta - 16 \sin \theta \cos \theta \). Here, each term is a product of \( \sin \theta \) and \( \cos \theta \).

To simplify such expressions, we focus on finding common patterns or terms. A common technique involves identifying like terms, which are parts of the expression that contain the same trigonometric functions. In our example, all terms include the combination \( \sin \theta \cos \theta \). This similarity helps us combine them more easily. Recognizing these patterns simplifies complex trigonometric expressions, a crucial skill in precalculus and calculus.

Simplifying Expressions

Simplifying expressions is about reducing their complexity without changing their value. In the exercise, both algebraic and trigonometric expressions are combined based on their like terms. Here's how it works:

1. **Identify Like Terms:** Like terms are parts of an expression that have the same variable combination. For example, \( 10SC + 4SC - 16SC \) has similar terms because they all include \( SC \). Similarly, \( 10 \sin \theta \cos \theta \) and \( 4 \sin \theta \cos \theta \) share trigonometric functions.

2. **Combine Coefficients:** To simplify, add or subtract the coefficients (the numbers in front) of like terms. In the trigonometric expression, \( 10 + 4 - 16 = -2 \). Thus, the expression simplifies to \( -2 \sin \theta \cos \theta \).

Simplification helps us handle more manageable expressions, a vital part of problem-solving in advanced mathematics.

Precalculus Operations

Precalculus includes foundational skills needed for studying calculus, like manipulating expressions and understanding functions. Operations in precalculus often involve simplifying expressions and solving equations involving trigonometric identities.

Through exercises like the one discussed, students learn to break down complex expressions into simpler forms. Key operations include:

• Identifying like terms.
• Combining them to simplify expressions.

These operations are essential not only for calculus but also in real-world applications like physics and engineering, where trigonometric functions model periodic phenomena like waves or oscillations. Mastering these concepts in precalculus lays the groundwork for more advanced studies, enabling students to tackle diverse mathematical challenges.