Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \(1 - \sin \theta \cos \theta\).

Step 1: Identify the Components

Notice that the expression \(\frac{\sin^3 \theta + \cos^3 \theta}{\sin \theta + \cos \theta}\) seems to match a known algebraic identity for the sum of cubes.

Step 2: Apply Sum of Cubes Formula

Recall the sum of cubes identity: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). In this case, let \(a = \sin \theta\) and \(b = \cos \theta\). So we get:\[\sin^3 \theta + \cos^3 \theta = (\sin \theta + \cos \theta)(\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta)\].

Step 3: Simplify Using Trigonometric Identity

Substitute the trigonometric identity \(\sin^2 \theta + \cos^2 \theta = 1\) into the expression. Therefore, the expression \(\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta\) simplifies to:\[1 - \sin \theta \cos \theta\].

Step 4: Cancel Out Common Factors

Since \(\sin \theta + \cos \theta\) is a factor in both the numerator and denominator, it can be cancelled out. Therefore, the simplified expression is \(1 - \sin \theta \cos \theta\).

Step 5: Final Expression

Thus, the expression simplifies to \(1 - \sin \theta \cos \theta\).

Key Concepts

Understanding the Sum of Cubes

When approaching algebraic expressions, recognizing patterns, like the sum of cubes, can ease the simplification process. The sum of cubes is a particular identity used often in algebra and trigonometry. The formula for the sum of cubes is:

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

Here, the expression is split into two simpler expressions: one involving the sum \((a + b)\) and another with a mixture of squares and cross products \((a^2 - ab + b^2)\).
In the given exercise, this identity helps in breaking down complex trigonometric powers into more manageable parts. By substituting, \(a = \sin \theta\) and \(b = \cos \theta\), we can directly apply this identity and transform the given expression into a product of factors. Leveraging such identities can make mathematical expressions simpler and more comprehensible.

The Art of Simplification

Simplification in mathematics, especially in dealing with complex trigonometric identities, is key to finding a clearer expression. Simplifying expressions involves reducing them to their simplest form. It is done by identifying and canceling common terms or using known identities.
For the initial expression, recognizing that \(\sin^2 \theta + \cos^2 \theta = 1\) is crucial. This identity allows the trigonometric components inside the expression to be consolidated into simpler forms.

• Replace \(\sin^2 \theta + \cos^2 \theta\) with 1.
• Focus on reducing the cross products.
• Cancel similar terms in the numerator and denominator.

By performing these distinct steps, you derive a less complex result, improving both the understanding and efficiency of handling the expression.

Grasping Algebraic Identity

Algebraic identities are innate tools to verify or simplify algebraic equations. These are equations true for all values of their variables. For trigonometric expressions, like the one in our exercise, such identities play an integral role in reconfiguring and simplifying terms.

• The sum of cubes itself is an algebraic identity that aids in disassembling and then reconstructing expressions into solvable components.
• Understanding these recurrent patterns allows for the cancelation of common factors efficiently.

These identities act as familiar templates that can simplify problem-solving strategies. By recognizing specific patterns in trigonometric problems, students can apply these identities to streamline their calculations, leading to more straightforward solutions and a deeper understanding of trigonometric functions.