Problem 12 - 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. $$\cos ^{4} x \sin ^{2} x$$

Step-by-Step Solution

Answer

Use lowering powers to express \( \cos^4 x \sin^2 x \) as a polynomial in \( \cos(2x) \).

Step 1: Recognize Common Identities

First, recall the trigonometric identities for lowering powers. We have:\[ \cos^2 x = \frac{1 + \cos(2x)}{2} \] and \[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]. We'll use these identities to express \( \cos^4 x \sin^2 x \) in terms of the first power of cosine.

Step 2: Lower the Power of Cosine

Replace \( \cos^4 x \) using the identity for \( \cos^2 x \):\[ \cos^4 x = (\cos^2 x)^2 = \left(\frac{1 + \cos(2x)}{2}\right)^2 \]This step helps simplify the expression to a form involving cosines.

Step 3: Lower the Power of Sine

Now replace \( \sin^2 x \) using its identity:\[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]Substituting this into the original expression gives:\[ \cos^4 x \sin^2 x = \left(\frac{1 + \cos(2x)}{2}\right)^2 \cdot \frac{1 - \cos(2x)}{2} \]

Step 4: Simplify the Expression

Expand the expression:\[ \cos^4 x \sin^2 x = \frac{(1 + \cos(2x))^2 (1 - \cos(2x))}{8} \]Simplify further using algebraic manipulation:\[ (1 + \cos(2x))^2 = 1 + 2\cos(2x) + \cos^2(2x) \]and\[ \cos^2(2x) = \frac{1 + \cos(4x)}{2} \].

Step 5: Combine and Simplify

Substitute \( \cos^2(2x) \):\[ \cos^4 x \sin^2 x = \frac{(1 + 2\cos(2x) + \frac{1 + \cos(4x)}{2})(1 - \cos(2x))}{8} \]Finally, distribute and collect like terms to write everything in terms of the first power of cosine.

Key Concepts

Power-Reducing Formulas

Power-reducing formulas are trigonometric identities that simplify expressions involving squared trigonometric functions. These formulas are very useful when trying to express higher powers of sine and cosine in terms of the first power.

Here are the most common power-reducing formulas:

• For cosine: \[ \cos^2 x = \frac{1 + \cos(2x)}{2} \]
• For sine: \[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]

These formulas essentially convert products or powers of trigonometric functions into forms that use double angle identities. This is particularly useful since expressions like \( \cos^4 x \sin^2 x \) can be daunting without simplifying.

In the process of simplifying trigonometric expressions, the power-reducing formulas are your first step, acting as the basic foundation upon which more involved simplifications, such as algebraic manipulation, are conducted.

Cosine

Cosine is one of the fundamental trigonometric functions and is essential in trigonometry. Its properties allow various transformations and simplifications.

When working with power-reducing formulas, cosine plays a crucial role. For instance, the formula \( \cos^2 x = \frac{1 + \cos(2x)}{2} \) shows how a second power of cosine can be rewritten in terms of a single cosine function. This is particularly useful in our original task, where we aim to express everything in terms of the first power of cosine.

Cosine functions appear frequently in trigonometric identities and are often used to condense or simplify expressions. It is important to be comfortable with cosine transformations when dealing with trigonometric identities.

Sine

Sine, similar to cosine, is a key trigonometric function that complements the use of other trigonometric identities. Its power-reducing formula helps in rewriting complex expressions.

In our context of power-reduction, \[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]plays a pivotal role. It allows for the transformation of squared sine terms into expressions involving cosine, simplifying complex expressions.

Understanding sine's capabilities and its relationship with cosine is fundamental. As seen in the simplification process, sine's power-reduction greatly aids in reducing expressions to simpler forms, contributing to a better understanding and easier computation of trigonometric problems. Grasping the foundational manipulation of sine is crucial for successful algebraic and trigonometric transformations.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying an expression using algebraic rules and properties. It is an essential step in working through trigonometric simplifications.

In our exercise, algebraic manipulation comes into play after applying power-reducing formulas. Once the trigonometric identities have been applied, expressions may still be complex. Algebraic manipulation involves expanding expressions, grouping, and simplifying terms.

For example:

• Expanding: \[ (1 + \cos(2x))^2 = 1 + 2\cos(2x) + \cos^2(2x) \]
• Substituting: Replace \( \cos^2(2x) \) with \( \frac{1 + \cos(4x)}{2} \).

These steps utilize algebraic manipulation to condense terms and express the initial complex expressions in much simpler, more manageable forms. This skill is essential for any student working with complex mathematics, particularly trigonometry.