Problem 1 - Precalculus with Calculus Previews

Question

In Problems \(1-12,\) use a product-to-sum formula in Theorem 4.7 .1 to write the given product as a sum of cosines or a sum of sines. $$ \cos 4 \theta \cos 3 \theta $$

Step-by-Step Solution

Answer

\( \cos 4\theta \cos 3\theta = \frac{1}{2} [ \cos 7\theta + \cos \theta ] \).

Step 1: Identify the Product-to-Sum Formula

To convert the product \( \cos A \cos B \) into a sum, use the product-to-sum formula: \( \cos A \cos B = \frac{1}{2} [\cos (A + B) + \cos (A - B)] \). Here, \( A = 4\theta \) and \( B = 3\theta \).

Step 2: Apply the Formula

Substitute \( A = 4\theta \) and \( B = 3\theta \) into the formula: \[ \cos 4\theta \cos 3\theta = \frac{1}{2} \left[ \cos ((4\theta + 3\theta)) + \cos ((4\theta - 3\theta)) \right] \].

Step 3: Simplify the Expression

Calculate the sums and differences inside the cosine terms: \( (4\theta + 3\theta) = 7\theta \) and \( (4\theta - 3\theta) = \theta \). Thus, the expression becomes \[ \frac{1}{2} [ \cos 7\theta + \cos \theta ] \].

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in simplifying and manipulating expressions involving trigonometric functions like sine, cosine, and tangent. These identities are equations that hold true for all values of the involved variables, as long as they lie within the domain of the functions. The identities provide relationships that connect different trigonometric functions, making it easier to evaluate and simplify expressions that may initially look complicated.

One of the most useful sets of identities are the product-to-sum formulas. These help in converting products of trigonometric functions into sums or differences.

• This conversion simplifies the expressions, which can make solving equations more manageable.
• For example, the product-to-sum formula for cosine is: \( \cos A \cos B = \frac{1}{2} [\cos (A + B) + \cos (A - B)] \).

In this exercise, you are asked to convert \( \cos 4\theta \cos 3\theta \) using this handy identity. Understanding and being able to apply these identities is fundamental in trigonometry and can be helpful across different applications in mathematics and physics.

Cosine Function

The cosine function is one of the primary trigonometric functions, alongside sine and tangent. It is usually defined in the context of a right-angled triangle or the unit circle. In a right-angled triangle, cosine of an angle \( \theta \) is the ratio of the length of the adjacent side to the length of the hypotenuse.

When considering the unit circle, the cosine function represents the x-coordinate of a point where a line, forming an angle \( \theta \) with the positive x-axis, intersects the circle. The function is periodic with a period of \( 2\pi \), meaning it repeats its values in intervals of \( 2\pi \).

• It is an even function, meaning that \( \cos(-x) = \cos(x) \).
• The range of the cosine function is from -1 to 1, reflecting its maximum and minimum values.

The cosine function is critical in various mathematical contexts and real-life applications, such as describing waves, circular motion, and alternating currents in physics.

Trigonometric Simplification

Trigonometric simplification involves reducing complex trigonometric expressions into simpler forms, making them easier to work with. This process often involves using known identities, such as product-to-sum formulas, to transform products into the sum of functions, or vice-versa.

By converting a trigonometric product like \( \cos 4\theta \cos 3\theta \) into a sum \( \frac{1}{2} [\cos 7\theta + \cos \theta] \), you can simplify calculations and integrate trigonometric functions more easily.

• This can help to isolate variables in equations or to make integration straightforward.
• Trigonometric simplification is particularly useful in calculus, physics, and engineering for solving differential equations and analyzing periodic functions.

Mastering the skill of simplification allows for a deeper understanding of the relationships between different trigonometric functions and how they can be manipulated for practical problem-solving. Moreover, it enhances your mathematical agility and ability to think critically in complex situations.