Problem 49 - 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. \( \sin^2 2x \cos^2 2x \)

Step-by-Step Solution

Answer

The expression \( \sin^2 2x \cos^2 2x \) rewritten in terms of the first power of cosine is \( \frac{- \cos 8x}{8} \)

Step 1: Identify the formula

First off, look at the power-reducing formulas for sine and cosine. These formulas are \[ \sin^2 x = \frac{1 - \cos 2x}{2} \] and \[ \cos^2 x = \frac{1+ \cos 2x}{2} \]

Step 2: Apply the formulas

Now apply these power-reducing formulas to the given expression \(\sin^2 2x \cos^2 2x\). Substituting \(\sin^2 2x\) and \(\cos^2 2x\) with their power-reducing formulas, we get : \[ \sin^2 2x \cos^2 2x = \left( \frac{1 - \cos 4x}{2} \right) \left( \frac{1+ \cos 4x}{2} \right) \]

Step 3: Simplify the expression

Multiply out the terms to simplify further. This results in : \[ \frac{1 - (\cos^2 4x)}{4} \]

Step 4: Rewriting using the power-reducing formula

Now take out the square in \( \cos^2 4x \) using the power-reducing formula \( \cos^2 x = \frac{1+ \cos 2x}{2} \). We therefore get \[ \frac{1 - \frac{1 + \cos 8x}{2}}{4} = \frac{1 - 1 - \cos 8x}{8} \] which simplifies to \[ \frac{- \cos 8x}{8} \]

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics, especially in trigonometry. They enable us to simplify and manipulate mathematical expressions involving trigonometric functions. Trigonometric identities include relationships such as Pythagorean, angle sum, double angle, and power-reducing formulas.

The power-reducing formulas help reduce higher power trigonometric terms into expressions involving trigonometric functions of lesser powers. This is particularly useful when simplifying complex expressions. For instance:

• Power-reducing formula for sine: \(\sin^2 x = \frac{1 - \cos 2x}{2}\)
• Power-reducing formula for cosine: \(\cos^2 x = \frac{1 + \cos 2x}{2}\)

The above formulas allow us to transform terms like \(\sin^2\) and \(\cos^2\) into forms that make it easier to perform algebraic manipulations, facilitating solutions for problems involving these trigonometric functions.

Sine and Cosine

Sine and cosine are the foundational trigonometric functions. They are defined based on the ratios of the sides of a right triangle relative to its angles.

- **Sine** of an angle \(x\) in a right triangle is the ratio of the length of the opposite side to the hypotenuse: \(\sin x = \frac{\text{Opposite}}{\text{Hypotenuse}}\).
- **Cosine** of an angle \(x\) is the ratio of the length of the adjacent side to the hypotenuse: \(\cos x = \frac{\text{Adjacent}}{\text{Hypotenuse}}\).

These functions are periodic, and their values repeat in cycles of \(2\pi\) radians. They also have important properties such as:

• 0 to 1 values in their primary cycle.
• Used in various formulas for simplifying expressions.
• Central in defining other trigonometric functions and in solving trigonometric equations.

Knowing how to manipulate these functions is crucial, especially when using identities and formulas to simplify equations like those in the given solution.

Algebraic Manipulation

Algebraic manipulation involves rearranging and modifying expressions to make them easier to interpret or solve. This is an essential skill in mathematics, particularly useful in working with equations involving trigonometric functions.

In the process of solving the given exercise, algebraic manipulation was vital. After applying the power-reducing identities, the expression \(\sin^2 2x \cos^2 2x\) initially transformed using these identities become:

\[\sin^2 2x \cos^2 2x = \left( \frac{1 - \cos 4x}{2} \right)\left( \frac{1 + \cos 4x}{2} \right)\]

Breaking it down further requires multiplication and simplification:

• By expanding the terms, consolidate the expression into a simpler fraction.
• Once simplified, replace \(\cos^2 4x\) using the power-reducing formula again, obtaining a form entirely involving the first power of cosine.
• This manipulation requires clear understanding of factorization and simplification strategies.

Effective algebraic manipulation makes solving trigonometric expressions efficient, leading to clean solutions involving well-defined terms like \(\frac{-\cos 8x}{8}\).