Problem 3 - Precalculus

Question

Carry out the indicated operations. (a) \(4 C^{3} S-12 C^{3} S\) (b) \(4 \cos ^{3} \theta \sin \theta-12 \cos ^{3} \theta \sin \theta\)

Step-by-Step Solution

Answer

(a) \(-8 C^3 S\); (b) \(-8 \cos^3 \theta \sin \theta\)."

Step 1: Identify Common Factors

In both expressions, identify common factors. For (a) \(4 C^3 S - 12 C^3 S\), the common factor is \(C^3 S\). For (b) \(4 \cos^{3} \theta \sin \theta - 12 \cos^{3} \theta \sin \theta\), the common factor is \(\cos^{3} \theta \sin \theta\).

Step 2: Factor Out the Common Term

For expression (a), factor out \(C^3 S\): \[4 C^{3} S - 12 C^{3} S = C^{3} S (4 - 12)\] For expression (b), factor out \(\cos^{3} \theta \sin \theta\): \[4 \cos^{3} \theta \sin \theta - 12 \cos^{3} \theta \sin \theta = \cos^{3} \theta \sin \theta (4 - 12)\]

Step 3: Simplify the Expression Inside the Parentheses

Simplify the expression inside the parentheses for both expressions. For (a): \(4 - 12 = -8\) For (b): \(4 - 12 = -8\)

Step 4: Rewrite the Complete Factored Expression

Use the simplified result from Step 3 to rewrite each expression. (a) becomes \(C^3 S (-8) = -8 C^3 S\) (b) becomes \(\cos^3 \theta \sin \theta (-8) = -8 \cos^3 \theta \sin \theta\)

Key Concepts

Understanding Common Factors

When working with algebraic expressions, identifying common factors is a crucial first step in simplifying them. A common factor is a term that appears in each part of the expression. Finding this shared element allows you to simplify the expression more easily.
For example, in the algebraic expression \[4 C^{3} S - 12 C^{3} S\], the term \(C^{3} S\) appears in both parts. This term is the common factor. Similarly, in the trigonometric expression \[4 \cos^{3} \theta \sin \theta - 12 \cos^{3} \theta \sin \theta\], \(\cos^{3} \theta \sin \theta\) is the common factor.

To factor out the common factor, you can rewrite the expression as:

• \(C^{3} S (4 - 12)\) for the first expression,
• \(\cos^{3} \theta \sin \theta (4 - 12)\) for the second expression.

This results in expressions that are easier to handle since they involve simpler operations inside the parentheses.

Basic Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are always true for every value of the angle involved. They are essential tools in simplifying trigonometric expressions and solving trigonometric equations.
In our example, the expression uses the cosine and sine functions:

• \( \cos \theta \) is the trigonometric function representing the ratio of the adjacent side to the hypotenuse in a right triangle.
• \( \sin \theta \) is the trigonometric function representing the ratio of the opposite side to the hypotenuse.

Understanding these basic functions helps in recognizing structures that can be simplified using identities. For instance, if we identify within more complex expressions where these functions repeat similarly, we might employ identities like Pythagorean identities to further simplify or solve problems. However, in this case, our focus remains on recognizing and factoring common terms.

Simplifying Expressions through Factoring

Simplifying algebraic or trigonometric expressions by factoring involves reducing them to their simplest form. This process makes the expression easier to work with—particularly for solving equations or integrating them into larger calculations.
Once you have factored out the common term, as seen in the expressions

• \(C^{3} S (4 - 12)\)
• \(\cos^{3} \theta \sin \theta (4 - 12)\)

the next step is simplifying what's inside the parentheses. In our examples:
\[(4 - 12) = -8\]
Substituting back gives us:

• \(-8 C^3 S\)
• \(-8 \cos^3 \theta \sin \theta\)

Both these adjusted expressions are simpler and represent the same value as the original expressions but are more straightforward when working with further algebraic manipulations or calculations.
In general, to simplify expressions, always look for factors and simplify their coefficients, variables, or terms to reduce them to the simplest form.