Problem 48 - 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. \( \sin^2 x \cos^4 x \)

Step-by-Step Solution

Answer

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

Step 1: Identify the Power Reducing Formulas

The power-reducing formulas that will be used are: \( \sin^2 x = \frac{1 - \cos 2x}{2} \) and \( \cos^2 x = \frac{1 + \cos 2x}{2} \). These formulas are employed to convert the higher power of sinusoidal functions into terms of the first power of cosine.

Step 2: Apply the Power Reducing Formula to \( \sin^2 x \)

Substitute the power-reducing formula into \( \sin^2 x \). This gives us \( \sin^2 x = \frac{1 - \cos 2x}{2} \).

Step 3: Apply the Power Reducing Formula to \( \cos^4 x \)

Substitute the power-reducing formula into \( \cos^4 x \). This simplifies to \( \cos^4 x = \frac{1 + \cos 2x}{2} \times \frac{1 + \cos 2x}{2} = \frac{1 + 2\cos 2x + \cos^2 2x}{4} \).

Step 4: Combine The Results

To obtain the final result, multiply the two results together: \( \sin^2 x \times \cos^4 x = \frac{1 - \cos 2x}{2} \times \frac{1 + 2\cos 2x + \cos^2 2x}{4} \). This is the original equation in terms of the first power of the cosine.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any values of the involved variables. These identities are useful tools in mathematics, particularly in calculus and trigonometry, where they assist in simplifying complex expressions.
Among these identities, some of the most well-known are the Pythagorean identities, angle sum and difference identities, double angle identities, and of course, the power-reducing formulas.

Importance of Power-Reducing Formulas

Power-reducing formulas are a subset of trigonometric identities that help in rewriting expressions with powers greater than one into a simpler cosine or sine format. They are particularly valuable when dealing with integrals or simplifying trigonometric expressions for further mathematical manipulation.
Using these identities, expressions with high powers of trig functions can be written in terms of first powers, easing the process of solving equations or evaluating integrals.

Cosine Functions

Cosine is one of the primary trigonometric functions, alongside sine and tangent. In the context of these exercises, cosine functions play a crucial role, especially when applying power-reducing formulas.

Understanding Cosine

Cosine of an angle is the ratio of the adjacent side to the hypotenuse in a right-angled triangle. On the unit circle, it represents the x-coordinate of a point corresponding to a specific angle. Cosine values vary between -1 and 1 as the angle moves around the circle.

Cosine in Power-Reducing Formulas

The power-reducing formula for \(\cos^2 x\) is \(\frac{1 + \cos 2x}{2}\). This expression rearranges \(\cos^2 x\) into a function solely of cosine with a double angle, simplifying the variable expression. Cosine functions play a central role since many trigonometric problems, like the given exercise, require converting higher power trigonometric terms to cosines for simplicity.

Higher Power Reduction

Higher power reduction involves rewriting expressions with powers of trigonometric functions into terms with lower powers. This is beneficial for simplifying expressions and solving equations in trigonometry more easily.

Simplifying Using Power Reduction

To simplify an expression like \(\sin^2 x \cos^4 x\) using power-reducing formulas, both sine and cosine components need to be transformed.

• Firstly, the power-reducing formula for \(\sin^2 x\) is applied: \(\sin^2 x = \frac{1 - \cos 2x}{2}\).
• Next, reduce \(\cos^4 x\) using its respective formula: \(\cos^4 x = \left(\frac{1 + \cos 2x}{2}\right)^2\). This further breaks down into simpler cosine terms.

By combining these results, the expression \(\sin^2 x \cos^4 x\) is written in terms of reduced powers of cosine, thus achieving a more manageable expression for mathematical operations.