Problem 152 - Precalculus

Question

For the following exercises, algebraically find an equivalent function, only in terms of \(\sin x\) and or \(\cos x\) , and then check the answer by graphing both equations. $$ \cos (4 x) $$

Step-by-Step Solution

Answer

\( \cos(4x) = 8\cos^4(x) - 8\cos^2(x) + 1 \).

Step 1: Understanding the problem

We need to express \( \cos(4x) \) in terms of \( \sin x \) and \( \cos x \). This involves using angle expansion formulas to break down \( 4x \) into simpler expressions that utilize \( \sin x \) and \( \cos x \).

Step 2: Use the Double Angle Formula

First, note that \( \cos(4x) = \cos(2(2x)) \). We can use the double angle formula for cosine: \( \cos(2\theta) = 2\cos^2(\theta) - 1 \). Apply this formula to get: \( \cos(4x) = 2\cos^2(2x) - 1 \).

Step 3: Apply Double Angle Formula Again

Now, apply the double angle formula again for \( \cos(2x) \): \( \cos(2x) = 2\cos^2(x) - 1 \), giving us \( \cos(4x) = 2(2\cos^2(x) - 1)^2 - 1 \).

Step 4: Expand and Simplify

Expand \((2\cos^2(x) - 1)^2\): \((2\cos^2(x) - 1)^2 = 4\cos^4(x) - 4\cos^2(x) + 1\). Substitute to get: \( \cos(4x) = 2(4\cos^4(x) - 4\cos^2(x) + 1) - 1 \). Simplify to \( \cos(4x) = 8\cos^4(x) - 8\cos^2(x) + 2 - 1 \). Hence, \( \cos(4x) = 8\cos^4(x) - 8\cos^2(x) + 1 \).

Step 5: Convert Sin to Cos

Use \( \sin^2(x) = 1 - \cos^2(x) \) if needed to express in terms of \( \sin x \). Here, we've expressed in terms of \( \cos x \), so this additional step isn't strictly necessary but could be used if the problem required. You can double-check by substituting back forms interchangeably if required.

Key Concepts

Angle Expansion

Angle expansion is a core concept in trigonometry that helps break down complex angles into simpler terms. This technique is particularly helpful when we encounter expressions such as \( \cos(4x) \) or \( \sin(4x) \) and need to express them in a more manageable form.
Understanding angle expansion is crucial because it allows us to transform higher angles into sums or multiples of smaller angles. It mainly involves using trigonometric identities like sum and difference formulas. In this context, for expressing \( \cos(4x)\), we can break it down using the double angle formula iteratively. In simpler terms, we find expressions for \( \cos(2x) \) and \( \, \cos(x) \, \) to get to \( \cos(4x) \).

• The main goal is to find an equivalent expression in terms of simpler trigonometric functions like \( \sin x \) and \( \cos x \).
• This approach simplifies graphing and allows for easier computation.
• Angle expansion is not limited to cosines and can be applied to sines and tangents as well.

Mastering angle expansion equips students with the ability to solve complex trigonometric problems more efficiently.

Double Angle Formula

The double angle formula is a specific trigonometric identity used to express trigonometric functions of twice an angle in terms of functions of the original angle. Specifically, for the cosine function, the formula is \( \cos(2\theta) = 2\cos^2(\theta) - 1 \).
Utilizing the double angle formula is essential when you need to simplify expressions such as \( \cos(4x) \) because it allows you to break it down step by step.

• First, recognize that \( \cos(4x) = \cos(2(2x)) \).
• Apply the double angle formula to find: \( \cos(4x) = 2\cos^2(2x) - 1 \).
• Then, apply it again to further expand: \( \cos(2x) = 2\cos^2(x) - 1 \).

Once you expand \( \cos(2x) \) to \( \cos(x) \), continue the process until everything is expressed in basic trigonometric terms, facilitating a clearer understanding of the equations. Using this technique simplifies computations and helps verify the results by making graphing errors less likely.

Cosine Function

The cosine function, a fundamental trigonometric function, is essential to understanding angles and periodic behavior in mathematics. The cosine function relates the adjacent side of a right triangle to its hypotenuse and is commonly denoted as \( \cos(\theta) \).
For expressions like \( \cos(4x) \), understanding the behavior of the cosine function helps transform and simplify the problem.

• The cosine function has a periodic cycle of \( 2\pi \), which means it repeats its values every \( 2\pi \).
• A key property is that \( \cos(\theta) = \cos(-\theta)\), indicating its even nature.
• The function's graph is a symmetric wave oscillating between -1 and 1, making it useful in modeling periodic phenomena.

The transformation from \( \cos(4x) \) to basic cosine terms involves using its properties and relationships, such as converting between \( \sin^2(x) \) and \( \cos^2(x) \). Understanding these fundamental aspects of the cosine function allows for efficient manipulation and application in various problems.