Problem 11 - Algebra and Trigonometry

Question

9–14 Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine, as in Example 4. $$\cos ^{2} x \sin ^{4} x$$

Step-by-Step Solution

Answer

The expression is \( \cos^2 x - 2\cos^4 x + \cos^6 x \).

Step 1: Identify the Given Expression

The given expression is \( \cos^{2} x \sin^{4} x \). Our goal is to rewrite this in terms of the first power of cosine.

Step 2: Use Trigonometric Identity to Lower Powers

Recall the identity \( \sin^2 x = 1 - \cos^2 x \). Use this identity to express \( \sin^4 x \) in terms of \( \cos \).

Step 3: Rewriting \(\sin^4 x \)

Since \( \sin^4 x = (\sin^2 x)^2 = (1 - \cos^2 x)^2 \), substitute this into the expression to get \( \cos^2 x (1 - \cos^2 x)^2 \).

Step 4: Expand the Expression

Expand the expression \((1 - \cos^2 x)^2\) to \(1 - 2\cos^2 x + \cos^4 x\) and substitute back into the expression to get \( \cos^2 x (1 - 2\cos^2 x + \cos^4 x) \).

Step 5: Distribute \( \cos^2 x \)

Distribute \( \cos^2 x \) across each term inside the parentheses: \[ \cos^2 x \cdot 1 - \cos^2 x \cdot 2\cos^2 x + \cos^2 x \cdot \cos^4 x = \cos^2 x - 2\cos^4 x + \cos^6 x \].

Step 6: Final Expression

The expression \( \cos^2 x - 2\cos^4 x + \cos^6 x \) is in terms of the first power of cosine.

Key Concepts

Lowering Powers in Trigonometry

Lowering powers is an essential technique used in trigonometry. It involves using trigonometric identities to reduce the power of trigonometric functions, making the expression simpler to work with. This technique is particularly useful when dealing with integrals or simplifying complex expressions.
To lower the powers of trigonometric functions such as sine or cosine, you can utilize well-known identities:

• The power-reducing formula for sine is: \[ \sin^2 x = \frac{1 - \cos 2x}{2} \]

In our specific exercise, the identity used is \( \sin^2 x = 1 - \cos^2 x \). By substituting higher powers of sine with expressions involving cosine, we effectively "lower" the power of the trigonometric expression. This simplification is highly beneficial in finding solutions or insights into the properties of functions.

Understanding Trigonometric Expressions

Trigonometric expressions consist of functions like sine, cosine, tangent, and their compositions. Simplifying these expressions often requires recognizing and applying various trigonometric identities. These are mathematical properties that express the equivalence between different trigonometric functions.
In the given problem, the expression \( \cos^2 x \sin^4 x \) is a combination of cosine and sine raised to powers. The task is to rewrite the entire expression in terms of the first power of cosine. By identifying such opportunities for simplification, you enable more efficient calculation or deeper analysis later.

• Breaking down the expression step-by-step, as shown in the solution, helps in not missing any critical identity that could simplify the problem further.

This thoughtful exploration of the trigonometric expression is crucial for successfully "lowering" powers and making interpretation straightforward.

The Role of the Cosine Function

Cosine is one of the fundamental trigonometric functions, integral to the study of angles and their relationships. The cosine function shows up frequently due to its natural properties and applications in various mathematical problems. Understanding how to manipulate cosine, especially using identities, is vital in solving complex expressions.
The goal in the provided solution is to express a trigonometric identity purely in terms of cosine. By doing so, the expression becomes simpler and more manageable. This is especially useful in calculus, where integration and differentiation are performed.

• In the problem, we began with \( \cos^2 x \sin^4 x \) and used power-reducing techniques to rewrite sine in terms of \( \cos x \).
• This allowed us to convert the entire expression solely in terms of the cosine function: \( \cos^2 x - 2\cos^4 x + \cos^6 x \).

Managing expressions with cosine leads to a more elementary understanding of the broader behavior of trigonometric functions in equations.