Problem 39 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{\sin ^{3} \theta+\cos ^{3} \theta}{\sin \theta+\cos \theta}$$

Step-by-Step Solution

Answer

The simplified expression is \( 1 - \sin \theta \cos \theta \).

Step 1: Identify a Suitable Formula

Given the expression is \( \frac{\sin ^{3} \theta + \cos ^{3} \theta}{\sin \theta + \cos \theta} \), it's helpful to recognize the formula for factoring a sum of cubes: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). We'll use this to simplify the numerator.

Step 2: Apply Sum of Cubes Formula

Apply the sum of cubes formula to the numerator by letting \( a = \sin \theta \) and \( b = \cos \theta \). This gives us \( \sin^3 \theta + \cos^3 \theta = (\sin \theta + \cos \theta)(\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta) \).

Step 3: Cancel Common Factors

The expression now becomes \( \frac{(\sin \theta + \cos \theta)(\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta)}{\sin \theta + \cos \theta} \). Since \( \sin \theta + \cos \theta \) is a common factor in both the numerator and the denominator, we can cancel it out.

Step 4: Simplify Remaining Expression

After cancelling the common factor, we are left with \( \sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta \). Simplify it using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). This gives us \( 1 - \sin \theta \cos \theta \).

Key Concepts

Sum of Cubes Formula

The sum of cubes formula is vital for simplifying expressions involving cube terms. When you see an expression that looks like a cube sum, such as \( a^3 + b^3 \), you can use this formula:

• \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)

This formula helps in breaking down the cubes into more manageable parts by factoring them into a product of a binomial and a trinomial.
Consider the exercise with trigonometric expressions. By letting \( a = \sin \theta \) and \( b = \cos \theta \), the sum of cubes formula can transform the expression \( \sin^3 \theta + \cos^3 \theta \) into \( (\sin \theta + \cos \theta)(\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta) \).
This process allows us to identify common factors and simplify the expression more easily. Recognizing and applying the sum of cubes formula is a powerful tool in algebra and beyond.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any value of the variables involved.
They are crucial in simplifying trigonometric expressions, solving trigonometric equations, and proving other mathematical statements. One of the most known identities is the Pythagorean identity:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

In our exercise, after factoring and cancelling out the common term, we end up with the expression \( \sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta \).
Using the Pythagorean identity here, we can substitute \( \sin^2 \theta + \cos^2 \theta \) with 1, simplifying the expression further to \( 1 - \sin \theta \cos \theta \).
Understanding and using trigonometric identities can significantly ease the process of simplifying complex trigonometric expressions.

Factoring Techniques

Factoring techniques are methods used to break down expressions into their simpler, "factor" components.
These techniques are pivotal in solving equations and simplifying expressions to make them more manageable. In the context of the given problem, the key step involves recognizing a sum of cubes and appropriately applying the sum of cubes formula.

• Identify the expression, \( a^3 + b^3 \), and apply the formula.
• Look for common factors between the numerator and the denominator to cancel them out.

The crucial technique used here is factoring by recognizing the sum of cubes, followed by cancelling the common factor \( \sin \theta + \cos \theta \) from both the numerator and the denominator.
Mastering these factoring techniques allows students to navigate and simplify mathematical problems with greater ease and efficiency.