Problem 44 - Precalculus with Limits

Question

Exercises 43-52, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. \( \sin^4 2x \)

Step-by-Step Solution

Answer

The expression \( \sin^4 2x \) rewritten in terms of the first power of cosine, using power-reducing formulas, is \( \frac{1}{4} - \frac{1}{2} \cos(4x) + \frac{1}{4} \cos^2(4x) \)

Step 1: Express in terms of Sine Square

Express \( \sin^4 2x \) as \( (\sin^2 2x)^2 \)

Step 2: Apply Power-Reducing Formula

Use the power-reducing formula: \(\sin^2 x = \frac{1 - \cos(2x)}{2}\). Perform this substitution to convert \( \sin^2 x \) into cosine: \( (\sin^2 2x)^2 = \left( \frac{1 - \cos(4x)}{2} \right)^2 \)

Step 3: Simplify the Expression

Simplify the resulting expression by expanding \( \left( \frac{1 - \cos(4x)}{2} \right)^2 \) and get: \( \sin^4 2x = \frac{1}{4} - \frac{1}{2} \cos(4x) + \frac{1}{4} \cos^2(4x) \)

Key Concepts

Power-Reducing Formulas

Power-reducing formulas are handy tools in trigonometry that help transform expressions with higher powers of sine or cosine functions into expressions with first powers. This is particularly useful in simplifying complex trigonometric expressions, especially when dealing with integrals and trigonometric identities.
The power-reducing formula for sine is given by:\[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]Similarly, the power-reducing formula for cosine is:\[ \cos^2 x = \frac{1 + \cos(2x)}{2} \]These formulas are derived from the double-angle identities and are extremely useful in calculus and pre-calculus courses. They convert quadratic expressions into linear ones, thus reducing the complexity of problem-solving. By applying these formulas, one can easily re-express trigonometric functions, aiding in various mathematical calculations and derivations.

• Helps simplify trigonometric expressions.
• Aids in solving trigonometric equations and integrals.
• Derived from double-angle identities.

Cosine Functions

The cosine function, often represented as \( \cos x \), is one of the fundamental trigonometric functions. It describes the relationship between the angle of a right-angled triangle and the length of the adjacent side over the hypotenuse. Cosine functions appear frequently in various mathematical contexts, particularly in periodic functions.
In the exercise, substituting sine with cosine using power-reducing formulas was necessary because cosine terms are often easier to handle. The transformation from \( \sin^4 2x \) to terms involving cosine illustrates their flexibility and ease of use in these situations.
Cosine functions have several key properties:

• Periodic with a period of \( 2\pi \).
• Symmetrical about the y-axis, making them even functions.
• Maximum value is 1 and minimum value is -1.
• Widely used in harmonic motion and wave equations.

Understanding how to manipulate expressions using cosine functions is crucial for mastering trigonometry and calculus. It allows for more rapid and accurate problem solving in both theoretical and applied contexts.

Trigonometric Expressions

Trigonometric expressions are formulas or equations that involve trigonometric functions like sine, cosine, tangent, and their reciprocals. These expressions are foundational to trigonometry and are used to model real-world phenomena such as waves and oscillations.
Rewriting and simplifying trigonometric expressions, like converting \( \sin^4 2x \) into cosine terms, is an essential skill. This often requires using identities such as power-reducing formulas, which rely on understanding these expressions' fundamental properties.
The goal of transforming trigonometric expressions:

• To simplify mathematical computations.
• To make complex problems more manageable.
• To reveal underlying patterns and symmetries.

Through practice, gaining proficiency in manipulating these expressions prepares students for more advanced studies in mathematics, engineering, and physics, where these concepts often play pivotal roles in solving real-world problems.