Problem 14 - Algebra and Trigonometry

Question

Simplify the trigonometric expression. $$ \cos ^{3} x+\sin ^{2} x \cos x $$

Step-by-Step Solution

Answer

The simplified expression is \(\cos x\).

Step 1: Factor Out Common Terms

Notice that both terms of the expression share a common factor: \(\cos x\). We can factor \(\cos x\) out of the terms: \[\cos^3 x + \sin^2 x \cos x = \cos x (\cos^2 x + \sin^2 x).\]

Step 2: Apply Pythagorean Identity

The expression inside the parentheses, \(\cos^2 x + \sin^2 x\), is a well-known Pythagorean identity: \[\cos^2 x + \sin^2 x = 1.\] Substitute the identity into the expression: \[\cos x (\cos^2 x + \sin^2 x) = \cos x \cdot 1 = \cos x.\]

Step 3: Simplify the Expression

After applying the identity and simplifying, the entire expression reduces to: \[\cos x.\]

Key Concepts

Factoring Trigonometric Expressions

Factoring trigonometric expressions is similar to factoring algebraic expressions. The goal is to find common factors in all terms of the expression, making it simpler to work with. In our exercise, the expression \( \cos^3 x + \sin^2 x \cos x \) has a common factor of \( \cos x \).

By factoring \( \cos x \) out of both terms, we rewrite the expression as \( \cos x (\cos^2 x + \sin^2 x) \). This is an essential step because it reduces the complexity of the expression and sets the stage for applying further trigonometric identities.

Always look for common trigonometric factors in expressions. This can often lead to significant simplifications. Recognizing and extracting common factors is a fundamental skill in simplifying complex trigonometric problems.

Pythagorean Identity

The Pythagorean identity is one of the cornerstone identities in trigonometry. It states that for any angle \( x \), the equation \( \cos^2 x + \sin^2 x = 1 \) holds true. This identity is a direct consequence of the Pythagorean theorem applied to the unit circle, where the radius is 1.

In our step-by-step solution, after factoring out \( \cos x \), we are left with \( \cos^2 x + \sin^2 x \) inside the parentheses. Applying the Pythagorean identity here is straightforward because it simplifies this part of the expression to 1. Thus, our expression becomes \( \cos x \cdot 1 = \cos x \).

Understanding and recalling the Pythagorean identity is crucial for solving various trigonometric problems efficiently. It helps in recognizing patterns and leads to easy simplification of otherwise complicated expressions.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves reducing them to their simplest form using known identities and algebraic manipulation. The aim is to make expressions easier to work with, whether in solving equations or in further mathematical analysis.

In this particular exercise, by factoring out \( \cos x \) and applying the Pythagorean identity, we reduce the original expression \( \cos ^{3} x+\sin ^{2} x \cos x \) to \( \cos x \).

Key steps in simplifying trigonometric expressions include:

• Looking for common factors to factor out.
• Applying trigonometric identities, like the Pythagorean identity or angle sum formulas.
• Simplifying the resulting expression to its simplest form.

By following these steps, complex expressions can often be turned into simple, easily interpretable forms. This not only aids in calculations but also deepens understanding of trigonometric relationships.