Problem 31 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

\( \cos^4 \frac{\theta}{2} = \frac{3 + 4\cos \theta + \cos (2\theta)}{8} \)

Step 1: Understanding the Expression

We start with the expression \( \cos^4 \frac{\theta}{2} \) and we need to simplify it using only a cosine function with exponent 1.

Step 2: Use the Power-Reduction Formula

We use the power-reduction formula: \( \cos^2 x = \frac{1 + \cos 2x}{2} \). For our expression, substitute \( x = \frac{\theta}{2} \), so \( \cos^2 \frac{\theta}{2} = \frac{1 + \cos \theta}{2} \).

Step 3: Reapply the Power-Reduction Formula

Since we have \( \cos^4 \frac{\theta}{2} = (\cos^2 \frac{\theta}{2})^2 \), substitute \( \cos^2 \frac{\theta}{2} = \frac{1 + \cos \theta}{2} \) into the squared expression to get \( \left( \frac{1 + \cos \theta}{2} \right)^2 \).

Step 4: Simplify the Fractions

Expand the squared term: \( \left( \frac{1 + \cos \theta}{2} \right)^2 = \frac{(1 + \cos \theta)^2}{4} \).

Step 5: Expand and Simplify the Polynomial

Expand the numerator: \( (1 + \cos \theta)^2 = 1 + 2\cos \theta + \cos^2 \theta \). Substitute \( \cos^2 \theta = \frac{1 + \cos (2\theta)}{2} \) into it, giving \( 1 + 2\cos \theta + \frac{1 + \cos (2\theta)}{2} \). Combine these terms to get \( \frac{2 + 4\cos\theta +1 + \cos (2\theta)}{2} \).

Step 6: Final Simplified Expression

Simplify the expression: \( 2 + 4\cos \theta + 1 + \cos(2\theta) = 3 + 4\cos \theta + \cos (2\theta) \). The final simplified expression for \( \cos^4 \frac{\theta}{2} \) is \( \frac{3 + 4\cos \theta + \cos (2\theta)}{8} \).

Step 7: Conclusion

Thus, \( \cos^4 \frac{\theta}{2} \) expressed in terms of the cosine function with exponent 1 is \( \frac{3 + 4\cos \theta + \cos (2\theta)}{8} \).

Key Concepts

Power-Reduction Formula

In trigonometry, the power-reduction formula is a handy tool that makes complex expressions simpler. This formula is especially useful when you encounter powers of trigonometric functions. Essentially, it transforms powers into expressions involving only the first power of the sine or cosine function. For the cosine function, the formula is:

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

To employ this formula, first identify the angle you're working with. In our example, we started with \( \cos^4 \frac{\theta}{2} \). By recognizing that \( \cos^4 \frac{\theta}{2} \) can be seen as \((\cos^2 \frac{\theta}{2})^2 \), we can simplify its squared component directly using the power-reduction formula. Applying the formula by substituting \( x = \frac{\theta}{2} \), gives us \( \cos^2 \frac{\theta}{2} = \frac{1 + \cos \theta}{2} \). Through this reduction, complicated expressions become more manageable, allowing further simplification.

Cosine Function

The cosine function is one of the fundamental trigonometric functions and is denoted by \( \cos \). It describes the relationship between the angles and lengths of a right triangle. Cosine has several noteworthy properties:

• It is an even function, meaning \( \cos(-x) = \cos(x) \).
• The value of the cosine function varies between -1 and 1 inclusive.
• Its graph is a wave that starts at 1, descends to -1, and returns to 1.

In trigonometric identities, the cosine function often appears in expressions that differ slightly from its base form. Consequently, simplifications frequently involve reconciling these forms back to standard cosine terms. Knowing these core properties helps understand its behavior and how it contributes to building and simplifying other expressions, as demonstrated when taking \( \cos^4 \frac{\theta}{2} \) and expressing it in terms of cosine with exponent 1.

Simplifying Expressions

Simplifying trigonometric expressions involves converting them into their simplest form. This process often requires the use of trigonometric identities to rewrite terms in less complex forms. In the given exercise, simplification was achieved by:

• Applying the power-reduction formula to lower the power of the cosine function.
• Expanding squared terms like \((1 + \cos \theta)^2\) into linear components.
• Combining like terms, with careful attention to getting a common denominator.

By expressing the expanded form as \(\frac{3 + 4\cos \theta + \cos (2\theta)}{8}\), all terms were brought together under a unified expression involving only the cosine function with exponent 1. This method reduces complications and is crucial for finding clearer, more precise solutions. Through these steps, one can navigate between complex trigonometric expressions and simpler, functionally equivalent forms, improving both understanding and computation efficiency.