Problem 43 - 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. \( \cos^4 x \)

Step-by-Step Solution

Answer

The expression \( \cos^4 x \) in terms of the first power of cosine using the power-reducing formulas is \( \cos^4 x = \frac{3}{8} + \frac{1}{2}\cos2x + \frac{1}{8}\cos 4x \)

Step 1: Apply the power-reducing formula

Substitute \( \cos^2 x \) with the power-reducing formula in the given expression. So, \( \cos^4 x = \left(\frac{1 + \cos 2x}{2}\right)^2

Step 2: Simplify the expression

Simplify the expression obtained in the previous step to get it in the first power of cosine. We get: \( \cos^4 x = \frac{1}{4} + \frac{1}{2}\cos2x + \frac{1}{4}\cos^2 2x

Step 3: Apply the power-reducing formula again

The term \( \cos^2 2x \) in our expression is still not in the first power. Thus, we need to apply the power-reducing formula again. Hence, \( \cos^4 x = \frac{1}{4} + \frac{1}{2}cos2x + \frac{1}{4}\left(\frac{1 + \cos 4x}{2}\right)

Step 4: Simplify the expression

Finally, simplify the expression obtained in step 3 to get the expression in terms of the first power of cosine. After simplifying, we get: \( \cos^4 x = \frac{3}{8} + \frac{1}{2}\cos2x + \frac{1}{8}\cos 4x

Key Concepts

Understanding Trigonometric Identities

Trigonometric identities are equations that hold true for all values within their domain. They relate the angles and sides of triangles and are crucial for simplifying complex expressions. Power-reducing formulas are specific trigonometric identities used to express higher powers of trigonometric functions, like \(\cos^2 x\), in terms of the first power.
These formulas stem from double angle identities, which express functions of doubled angles. For cosine, the power-reducing formula allows us to break down expressions involving \(\cos^4 x\) into forms involving \(\cos 2x\) and \(\cos 4x\).

• Power-Reducing Formula for cosine: \(\cos^2 x = \frac{1 + \cos 2x}{2}\)
• Uses: Simplifying powers in trigonometric expressions effectively.

Understanding these identities is key to solving and simplifying a variety of mathematics problems efficiently.

Exploring the Cosine Function

The cosine function is one of the primary trigonometric functions, crucial in both pure and applied mathematics. It describes the x-coordinate of a point on the unit circle.
Cosine can appear in different forms:

• Basic Form: Represents angles directly related to a right triangle.
• Double Angle: Given by \(\cos 2x = 2\cos^2 x - 1\), useful for deriving other identities.

In our exercise, these forms help us express higher powers of cosine in simpler, more manageable terms. Understanding how to manipulate these forms is essential for advanced trigonometry problems.

Algebraic Manipulation Techniques

Algebraic manipulation involves reworking expressions using algebraic rules to simplify or solve them. This is an essential skill in mathematics, particularly when handling trigonometric identities.
When working with power-reducing formulas:

• Substitute: Replace \(\cos^2 x\) using its identity to simplify expressions like \(\cos^4 x\).
• Expand & Simplify: After substitution, expand the expression and combine like terms for clarity.

These steps reduce the problem to the first power of cosine, making the results easier to interpret. By applying these techniques, complex trigonometric problems become more accessible and straightforward to solve.