Problem 14 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \( \cos x \).

Step 1: Identify a common factor

Look at the expression \( \cos^3 x + \sin^2 x \cos x \). Notice that both terms contain \( \cos x \) as a factor. We can factor \( \cos x \) out of the expression.

Step 2: Factor out the common factor

Factor \( \cos x \) from each term to get \( \cos x ( \cos^2 x + \sin^2 x ) \).

Step 3: Recognize the Pythagorean Identity

Recall that the Pythagorean identity states \( \cos^2 x + \sin^2 x = 1 \). Use this identity to replace \( \cos^2 x + \sin^2 x \) with 1 in the expression.

Step 4: Simplify the expression

Substitute the identity into the expression: \( \cos x (1) = \cos x \). The expression simplifies to just \( \cos x \).

Key Concepts

Factoring Trigonometric Expressions

Factoring in trigonometry is just like factoring in algebra. It involves looking for common elements in each term of a trigonometric expression and grouping them together. This makes the expression simpler and easier to work with. Consider the example given in our problem:

• The expression is \( \cos^3 x + \sin^2 x \cos x \).
• Both terms, \( \cos^3 x \) and \( \sin^2 x \cos x \), share the common factor \( \cos x \).

We factor \( \cos x \) out from each term:

• What's left inside the parentheses after factoring out \( \cos x \) is \( \cos^2 x + \sin^2 x \).

By factoring, we have simplified the expression to \( \cos x ( \cos^2 x + \sin^2 x ) \). Factoring is an essential technique because it often sets the stage for further simplifications, such as using identities, which simplify the process even more.

Pythagorean Identity

The Pythagorean Identity is one of the fundamental identities in trigonometry. It states that for any angle \( x \), the sum of the squares of sine and cosine is always equal to one: \[ \cos^2 x + \sin^2 x = 1 \]This identity is incredibly powerful because it holds true for any angle and is used extensively in trigonometric simplification. Let's see how we apply this identity to our expression:

• After factoring, we get \( \cos x ( \cos^2 x + \sin^2 x ) \).
• According to the Pythagorean Identity, \( \cos^2 x + \sin^2 x \) can be replaced with 1.

So, the expression becomes \( \cos x \times 1 \), which simplifies directly to \( \cos x \). The Pythagorean Identity is an essential tool for simplifying many trigonometric expressions because it reduces products and sums to much simpler forms.

Trigonometric Simplification

Trigonometric simplification is about making trigonometric expressions as simple as possible. Simplifying can often help you see patterns or find solutions that weren't initially obvious. There are several strategies for simplifying, and it can involve:

• Factoring expressions, like in our example with \( \cos x \).
• Using trigonometric identities, such as the Pythagorean Identity.

For our expression, we simplified it by:

• Factoring out \( \cos x \), which gave us \( \cos x ( \cos^2 x + \sin^2 x ) \).
• Using the Pythagorean Identity to replace \( \cos^2 x + \sin^2 x \) with 1.

This sequence of steps provided the simplest form of the expression: \( \cos x \). Starting with what seemed like a complex expression, we used known patterns and identities to break it down to a basic form. Efficient simplification not only makes expressions more manageable but is often key in solving trigonometric equations.