Problem 88 - Precalculus with Limits

Question

In Exercises 81-90, use the product-to-sum formulas to write the product as a sum or difference. \( \cos 2 \theta \cos 4 \theta \)

Step-by-Step Solution

Answer

The product \( \cos 2 \theta \cos 4 \theta \) can be expressed as a sum using the product-to-sum formula, resulting in \( \cos 2 \theta \cos 4 \theta = 1/2 [\cos(2 \theta) + \cos(6 \theta)] \).

Step 1: Identify and Apply the Product-to-Sum Formula

For the given expression \( \cos 2 \theta \cos 4 \theta \), we can use the product-to-sum formula: \( \cos a \cos b = 1/2[\cos(a-b) + \cos(a+b)] \). Here a is \( 2 \theta \) and b is \( 4 \theta \). After substitution we obtain \( \cos 2 \theta \cos 4 \theta = 1/2 [\cos(2 \theta - 4 \theta) + \cos(2 \theta + 4 \theta)] \).

Step 2: Simplify the Expression

Now, we just need to simplify this expression, resulting in \( \cos 2 \theta \cos 4 \theta = 1/2 [\cos(-2 \theta) + \cos(6 \theta)] \). Also, considering the property \(\cos(-x) = \cos(x)\), this expression finally simplifies to \( \cos 2 \theta \cos 4 \theta = 1/2 [\cos(2 \theta) + \cos(6 \theta)] \).

Key Concepts

Trigonometric Identities

Trigonometric identities are like shortcuts that make working with trigonometric functions easier. They relate different trigonometric functions and expressions. Learning these identities can simplify solving equations and lower complexity in calculations.

There are several key trigonometric identities:

• Pythagorean Identities: These include \( \sin^2(x) + \cos^2(x) = 1 \).
• Angle Sum and Difference Identities: Like \( \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \).
• Product-to-Sum Formulas: These help you convert products of trigonometric functions into sums or differences, such as \( \cos a \cos b = \frac{1}{2} [\cos(a-b) + \cos(a+b)] \).

Using these identities effectively makes complex trigonometric expressions manageable and simplifies problem-solving.

Cosine Function

The cosine function is one of the primary trigonometric functions. It is written as \( \cos \theta \) where \( \theta \) is the angle. The cosine function indicates the x-coordinate of the chosen point on the unit circle for angle \( \theta \). Here are some important features of the cosine function:

• Range: Values of \( \cos \theta \) span from -1 to 1.
• Periodicity: Cosine functions repeat every \( 2\pi \) radians.
• Even Function: Since \( \cos(-\theta) = \cos \theta \), it has reflective symmetry about the y-axis.
• Key Values: \( \cos 0 = 1 \), \( \cos \frac{\pi}{2} = 0 \), \( \cos \pi = -1 \).

Cosine plays a vital role in product-to-sum formulas, especially in converting products into simpler sums, as seen in the original exercise with \( \cos 2 \theta \cos 4 \theta \).

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves using identities to make a complex expression more digestible. In our exercise context, this means transforming \( \cos 2 \theta \cos 4 \theta \) into a form that is easier to work with. This process is crucial:

• Reduces Complexity: By using identities, you change products into sums or differences, simplifying equations.
• Highlights Patterns: Simple forms can reveal patterns or solutions that are hard to see in their original complex form.
• Derivation: For \( \cos 2 \theta \cos 4 \theta \), using the product-to-sum formula leads to \( \frac{1}{2} [\cos(2\theta) + \cos(6\theta)] \).

This simplified expression highlights how knowing trigonometric identities, like the product-to-sum, can make otherwise challenging tasks much more straightforward.