Problem 3 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a sum or difference. $$ \cos 6 u \cos (-4 u) $$

Step-by-Step Solution

Answer

\( \cos 6u \cos (-4u) = \frac{1}{2} \cos 2u + \frac{1}{2} \cos 10u \)

Step 1: Use Product-to-Sum Formulas

To express the product \( \cos 6u \cos (-4u) \) as a sum or difference, we'll use the product-to-sum identity for cosine: \[ \cos A \cos B = \frac{1}{2} [\cos (A + B) + \cos (A - B)] \] where \( A = 6u \) and \( B = -4u \).

Step 2: Substitute Values into Formula

Substitute \( A = 6u \) and \( B = -4u \) into the product-to-sum formula: \[ \cos 6u \cos (-4u) = \frac{1}{2} [\cos (6u + (-4u)) + \cos (6u - (-4u))] \]

Step 3: Simplify Inside the Cosine

Calculate the expressions inside the cosine functions: - \( 6u + (-4u) = 2u \)- \( 6u - (-4u) = 10u \) Thus, we have: \[ \cos 6u \cos (-4u) = \frac{1}{2} [\cos 2u + \cos 10u] \]

Step 4: Write Final Expression as a Sum

The expression for \( \cos 6u \cos (-4u) \) in terms of a sum is: \[ \cos 6u \cos (-4u) = \frac{1}{2} \cos 2u + \frac{1}{2} \cos 10u \]

Key Concepts

Product-to-Sum Formulas

The product-to-sum formulas are trigonometric identities that allow expressions involving the products of sines and cosines to be rewritten as sums or differences of sines and cosines. These formulas simplify the multiplication of trigonometric functions into a more approachable form, aiding in integration, differentiation, and other mathematical applications.

For the cosine function, the product-to-sum formula is:

• \( \cos A \cos B = \frac{1}{2} [\cos (A + B) + \cos (A - B)] \)

This formula expresses that the product of two cosines can be converted into a sum of two cosine terms. By doing this, we transform a complex product into an easier trigonometric expression. This formula is incredibly useful when dealing with exercises like the one at hand, where we have \( \cos 6u \cos (-4u) \). It allows us to express this as a simple sum:

\( \cos 6u \cos (-4u) = \frac{1}{2} [\cos (2u) + \cos (10u)] \).

Using this formula facilitates calculations and helps when analyzing the behavior of trigonometric functions.

Cosine Function

The cosine function, one of the fundamental trigonometric functions, relates the angle of a right triangle to the ratios of its sides. More specifically, if you have a right triangle, the cosine of one of its angles is the length of the adjacent side divided by the length of the hypotenuse.

In a broader sense, cosine can be conceptualized as a wave, as it represents the periodic nature of circles and cycles in mathematics and engineering. This wave-like form is why the cosine function is prevalent in physics and engineering, where periodic phenomena like sound waves and electromagnetic waves are at play.

When we look at transformations like \( \cos 6u \) or \( \cos (-4u) \), they tell us how the basic cosine wave stretches, squeezes, and flips. Negative angles reflect the function across the vertical axis, but cosine enjoys symmetry; thus, \( \cos(-\theta) = \cos(\theta) \). This property simplifies many computations and was used in the given exercise to compute \( \cos 6u \cos (-4u) \).

Angle Addition and Subtraction

Angle addition and subtraction formulas offer a way to calculate trigonometric functions for specific angles as a result of known values. They are particularly useful for determining the sine, cosine, and tangent of the sum or difference of two angles.

The essence of these formulas, like \( \cos(A + B) \) and \( \cos(A - B) \), is foundational in trigonometric transformations and proves helpful in simplifying complex trigonometric expressions as seen in the exercise. For example:

• \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
• \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

These formulas were subtly applied behind the scenes in the transformation from product forms to sum forms in the exercise. They help not only in breaking down functions like \( \cos 6u + \cos 10u \) but also in understanding their behavior and characteristics.

Comprehending these formulae allows for deeper insight into both theoretical and practical applications, enhancing your problem-solving toolkit when dealing with trigonometric expressions.