Problem 45 - 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. \( \cos^4 2x \)

Step-by-Step Solution

Answer

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

Step 1: Identify the formula to use

The first step is to identify the correct power-reducing formulas for \(\cos^4 2x\). In this case, since the cosine is to the power of 4, we can break that down into \(\cos^2 2x * \cos^2 2x\). We will now use the formula for cosine which is \(\cos^2 A = \frac{1 + \cos 2A}{2}\).

Step 2: Apply the power-reducing formula

Now we can apply the formula to \(\cos^2 2x\). Let us now substitute \(A = 2x\) in the formula \(\cos^2 A = \frac{1 + \cos 2A}{2}\) to get \(\cos^2 2x = \frac{1 + \cos 4x}{2}\). We will apply this twice since we have \(\cos^2 2x * \cos^2 2x\) which gives us expression \(\left(\frac{1 + \cos 4x}{2}\right)\left(\frac{1 + \cos 4x}{2}\right)\).

Step 3: Simplify the expression

Now you can simplify the expression \(\left(\frac{1 + \cos 4x}{2}\right)\left(\frac{1 + \cos 4x}{2}\right)\), by multiplying and using distributive property, which gives \( \frac{1 + 2\cos 4x + \cos^2 4x}{4}\). Again, we will use the power-reducing formula to simplify the \(\cos^2 4x\). Using the formula \(\cos^2 A = \frac{1 + \cos 2A}{2}\), substituting \(A = 4x\), we get \(\cos^2 4x = \frac{1 + \cos 8x}{2}\). Substituting this back into the expression, we get the simplified expression to be \( \frac{1 + 2\cos 4x + \frac{1 + \cos 8x}{2}}{4}\). Simplify it further to get the final expression \( \frac{2 + 4\cos 4x + 1 + \cos 8x}{8}\) or \( \frac{3 + 4\cos 4x + \cos 8x}{8}.\)

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function, crucial for understanding waves, oscillations, and circles. On the unit circle, which is a circle with a radius of one centered at the origin of a coordinate system, the cosine of an angle represents the x-coordinate of the point where the terminal side of the angle intersects the circle. It is usually written as \( \cos(\theta) \) for an angle \( \theta \).

When dealing with the cosine function squared, such as \( \cos^2(\theta) \) or to higher powers, the notation signifies the cosine of an angle taken to that power, not the angle itself. Power-reducing formulas allow us to express these higher powers of cosine in terms of the first power, which is useful for simplifying trigonometric expressions during integration, solving equations, and analyzing wave forms.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a critical skill in both high school and college-level mathematics. These expressions, which involve trigonometric functions like sine, cosine, and tangent, often appear complex and intimidating at first. However, simplifying them makes the expressions more manageable and suitable for solving problems and proving identities.

Power-reducing formulas, like the one used in our exercise, are a particular kind of trigonometric identities that help transform higher even powers of sine and cosine to the first power. These formulas are derived from the double-angle formulas and are essential tools for simplifying expressions involving powers of trigonometric functions along with other techniques such as factoring, using Pythagorean identities, or even switching to different trigonometric functions.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved variables where both sides of the equation are defined. They are essential for simplifying trigonometric expressions and transforming them into forms that are more easily integrated or solved.

The power-reducing formula used in the exercise, \( \cos^2 A = \frac{1 + \cos 2A}{2} \), is one such identity. It is particularly useful as it reduces second powers to the first power by incorporating the double-angle identity. Mastering these identities allows students to manipulate complex trigonometric equations, proving relationships between angles and sides in geometry, and solving intricate calculus problems. By understanding these identities, students can approach trigonometric problems with a set of powerful tools to simplify and solve them.