Problem 9 - Algebra and Trigonometry

Question

9–14 Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine, as in Example 4. $$\sin ^{4} x$$

Step-by-Step Solution

Answer

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

Step 1: Recognize the Trigonometric Identity

We will start by recognizing that we need to use the power-reducing identities. The power-reducing formula for \( \sin^2 x \) is \( \sin^2 x = \frac{1 - \cos(2x)}{2} \).

Step 2: Apply the Power-Reducing Identity

Now, apply the \( \sin^2 x \) formula to \( \sin^4 x \): \( \sin^4 x = (\sin^2 x)^2 = \left( \frac{1 - \cos(2x)}{2} \right)^2 \).

Step 3: Simplify the Expression

Expand the expression: \( \left( \frac{1 - \cos(2x)}{2} \right)^2 = \frac{1 - 2\cos(2x) + \cos^2(2x)}{4} \).

Step 4: Use the Power-Reducing Identity Again

Replace \( \cos^2(2x) \) using its power-reducing identity: \( \cos^2(2x) = \frac{1 + \cos(4x)}{2} \).

Step 5: Substitute and Combine Fractions

Insert the expression for \( \cos^2(2x) \) into the equation: \[ \frac{1 - 2\cos(2x) + \frac{1 + \cos(4x)}{2}}{4} = \frac{2 - 4\cos(2x) + 1 + \cos(4x)}{8} \].

Step 6: Final Simplification

Combine and simplify the numerator: \( \frac{3 - 4\cos(2x) + \cos(4x)}{8} \). This expression is now in terms of the first power of cosine.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the involved variables. They play an important role in simplifying complex trigonometric expressions and solving equations. Power-reducing formulas are a specific type of trigonometric identity used to express higher powers of trigonometric functions, like sine and cosine, in terms of their first powers. This approach is especially important in calculus and higher mathematics. The power-reducing formula for sine is:

• \( \sin^2 x = \frac{1 - \cos(2x)}{2} \)

These formulas help convert an expression, like \( \sin^4 x \), into a simpler form that is easier to integrate or differentiate. Understanding how to use these identities is crucial for solving trigonometric problems more effectively. They help break down complex trigonometric forms into manageable parts.

First Power of Cosine

In trigonometry, expressing a function in terms of the first power of cosine means converting terms with higher powers into forms involving \( \cos x \) or \( \cos(2x) \), etc. This makes the expression simpler and easier to handle. To illustrate, if we have \( \sin^4 x \), we use the power-reducing formulas to express it in terms of \( \cos(2x) \). We first express \( \sin^2 x \) using its identity:

• \( \sin^2 x = \frac{1 - \cos(2x)}{2} \)

Then, we raise \( \sin^2 x \) to the power of 2 because we started with \( \sin^4 x \):

• \( \sin^4 x = \left( \frac{1 - \cos(2x)}{2} \right)^2 \)

By expanding and further simplifying, you can express the original function in terms of \( \cos \), utilized for integration or differentiation in more advanced problems.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves reducing them to their simplest form or a more manageable form for further analysis or calculation. Power-reducing identities, like those used to convert \( \sin^4 x \) into a sum of terms involving the first power of cosine, are perfect tools for this task. Let's look closer at the simplification process:

• Start by recognizing and using the relevant identities, such as applying \( \sin^2 x = \frac{1 - \cos(2x)}{2} \) to each squared sine function.
• Rewriting squared terms, so any higher power terms are expressed using basic trigonometric functions.
• Finally, combine like terms and reduce, aiming to have a clean expression, ideally including the lowest power functions of \( \cos \).

Simplifying expressions is especially useful in calculus and algebra, allowing for convenient integration, differentiation, and function manipulation. By breaking down complex expressions, students can better understand underlying trigonometric principles and solve intricate math problems more efficiently.