Problem 33 - Algebra and Trigonometry with Analytic Geometry

Question

Express in terms of the cosine function with exponent \(1 .\) $$ \sin ^{4} 2 x $$

Step-by-Step Solution

Answer

\( \sin^4 2x = \frac{3}{8} - \frac{1}{2}\cos 4x + \frac{1}{8} \cos 8x \).

Step 1: Use the Identity for Sine Squared

Recall the identity \( \sin^2 \theta = 1 - \cos^2 \theta \). Using this identity, we can express \( \sin^4 2x \) as \( (\sin^2 2x)^2 = (1 - \cos^2 2x)^2 \).

Step 2: Expand the Square

Now, expand the expression \( (1 - \cos^2 2x)^2 \) using the formula \((a-b)^2 = a^2 - 2ab + b^2\). This gives \( 1 - 2\cos^2 2x + \cos^4 2x \).

Step 3: Express Cosine to the Fourth Power

Notice that \( \cos^4 2x = (\cos^2 2x)^2 \). We can use the double angle identity \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \) to express \( \cos^2 2x \) in terms of \( \cos 4x \): \( \cos^2 2x = \frac{1 + \cos 4x}{2} \).

Step 4: Simplify Cosine to the Fourth Power

Write \( \cos^4 2x = \left(\frac{1 + \cos 4x}{2}\right)^2 \) which simplifies to \( \frac{1 + 2\cos 4x + \cos^2 4x}{4} \).

Step 5: Combine All Terms

Insert \( \cos^4 2x = \frac{1 + 2\cos 4x + \cos^2 4x}{4} \) into the expanded expression: \( 1 - 2\cos^2 2x + \Bigg(\frac{1 + 2\cos 4x + \cos^2 4x}{4}\Bigg) \).

Step 6: Simplify the Expression Completely

Combine all the terms to express the entire polynomial in terms of powers of cosine to the first power. Simplifying might further require using \( \cos^2 4x = \frac{1 + \cos 8x}{2} \). After simplification, all terms should reflect expressions involving cosine only with power 1 or higher.

Key Concepts

Cosine Function

The cosine function, represented as \( \cos \theta \), is one of the fundamental trigonometric functions. It describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Cosine is periodic with a cycle of \( 2\pi \), and it ranges from -1 to 1.

Understanding the cosine function is crucial because it frequently appears in various trigonometric identities and equations. For instance, in solving trigonometric problems, converting one function to another can often simplify the solution process. A key property of the cosine function is its relationship with the sine function, particularly through identities which can simplify calculations.

This is important in simplifying expressions such as \( \sin^4 2x \) into terms only involving the cosine function. Using known identities, we can convert higher powers of sine into a combination of cosine functions, leading to simpler expressions.

Sine Function

The sine function, denoted as \( \sin \theta \), is another basic trigonometric function. It represents the ratio of the opposite side to the hypotenuse in a right triangle. Like cosine, sine is periodic with a period of \( 2\pi \), and it also ranges from -1 to 1.

Trigonometric identities often include transformations between sine and cosine, such as \( \sin^2 \theta = 1 - \cos^2 \theta \). In our example, expressing \( \sin^4 2x \) in terms of cosine requires applying this identity. By recognizing \( \sin^2 \theta \), we can convert the sine power to an expression involving \( \cos \theta \).

Understanding how sine interacts with cosine, particularly through these identities, allows us to manipulate and simplify trigonometric expressions. This is done by breaking down the powers or products of trigonometric functions into simpler forms.

Double Angle Identity

The double angle identities are powerful tools in trigonometry, essential for transforming trigonometric expressions. The core principle of a double angle identity lies in expressions like \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \) or \( \cos^2 \theta = \frac{1+\cos(2\theta)}{2} \).

For example, to simplify \( \cos^4 2x \), we utilize the identity \( \cos^2 2x = \frac{1 + \cos 4x}{2} \). This allows us to express the term involving \( \cos^4 2x \) in simpler terms involving expressions with \( 2x \) and \( 4x \).

Double angle identities are crucial for reducing the complexity of trigonometric expressions. They help us express trigonometric functions in various equivalent forms, often making the solution more straightforward.

Algebraic Manipulation

Algebraic manipulation involves rearranging, expanding, or factoring expressions to simplify or solve them. This process is vital in working with complex trigonometric identities.

When expressing \( \sin^4 2x \) in terms of cosine, expanding expressions like \((1 - \cos^2 2x)^2\) involves applying the distributive property \((a-b)^2 = a^2 - 2ab + b^2\). This results in an expanded polynomial of cosine terms.

Further combining terms involves inserting and simplifying expressions like \( \cos^4 2x = \left(\frac{1 + \cos 4x}{2}\right)^2 \). Through algebraic manipulations, we convert the trigonometric terms to reflect their cosine counterparts with manageable coefficients. Mastery of these techniques is indispensable for any student dealing with higher-level mathematics, allowing accurate and simpler expression transformations.