Problem 8 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a sum or difference. $$ 5 \cos u \cos 5 u $$

Step-by-Step Solution

Answer

\(\frac{5}{2} [\cos(6u) + \cos(4u)]\)

Step 1: Identify the Trigonometric Identity

Recognize that the expression \(5 \cos u \cos 5u\) involves the cosine function and can be rewritten using the product-to-sum identities. Specifically, the identity for the product of cosines is: \[ \cos A \cos B = \frac{1}{2} [\cos(A+B) + \cos(A-B)] \] We will use this identity to express the given expression as a sum.

Step 2: Substitute into the Identity

Apply the product-to-sum identity to \(\cos u \cos 5u\). Here, \(A = u\) and \(B = 5u\). Substitute these into the identity: \[ \cos u \cos 5u = \frac{1}{2} [\cos(u + 5u) + \cos(u - 5u)] \]which simplifies to: \[ \frac{1}{2} [\cos(6u) + \cos(-4u)] \]. Note that \(\cos(-4u) = \cos(4u)\) because the cosine function is even.

Step 3: Simplify the Expression

Now, simplify the expression using the fact that \(\cos(-4u) = \cos(4u)\):\[ \cos u \cos 5u = \frac{1}{2} [\cos(6u) + \cos(4u)] \]. Don't forget to multiply by 5, which was a coefficient in the original expression: \[ 5 \cos u \cos 5u = 5 \times \frac{1}{2} [\cos(6u) + \cos(4u)] = \frac{5}{2} [\cos(6u) + \cos(4u)] \].

Key Concepts

Product-to-Sum Identities

Trigonometric identities are powerful tools that simplify complex mathematical expressions. A particularly useful type is the product-to-sum identities. These identities allow you to rewrite products of trigonometric functions as sums or differences, making calculations easier.

When working with the cosine function, the product-to-sum identity is given by:

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

This equation is applied when you encounter a product of cosines. By expressing the product as a sum, you can simplify the expression, making complex problems more manageable.

In our example, we use it to simplify \(5 \cos u \cos 5u\) by shifting from multiplication to addition.

Cosine Function

The cosine function is one of the foundational elements in trigonometry. It is an even function and appears frequently in various mathematical contexts. The general cosine function, \(\cos(x)\), represents the adjacent side over the hypotenuse in a right triangle or the x-coordinate in the unit circle.

Its properties include periodicity and symmetry:

• Periodicity: The cosine function has a period of \(2\pi\), meaning it repeats every \(2\pi\) units.
• Symmetry: As an even function, cosine satisfies \(\cos(-x) = \cos(x)\).

These properties make the cosine function particularly useful in trigonometric identities, helping simplify and solve trigonometric expressions, as seen in the problem concerning \(\cos u \cos 5u\).

Even Function

Understanding even functions is essential for simplifying trigonometric expressions. Even functions are symmetric about the y-axis. This means that they satisfy the equation \(f(-x) = f(x)\) for every \(x\) in their domain.

The cosine function is a prime example of an even function. This property implies that when dealing with expressions like \(\cos(-4u)\), it can be easily rewritten as \(\cos(4u)\), simplifying the problem at hand.

Recognizing and applying this property helps streamline processes in mathematics, especially in trigonometric manipulations and identity transformations. This symmetry enables easier integration and manipulation when using product-to-sum identities.

Simplifying Expressions

Simplification is the heart of solving trigonometric problems. In many math problems, complex expressions can be daunting; thus, simplification makes them more approachable.

When simplifying expressions using identities, like the product-to-sum identities, it’s crucial to apply properties correctly and to note any function characteristics, such as the even nature of cosine.

• Apply identities to break down products into sums or differences.
• Use function properties to simplify terms further, as in converting \(\cos(-4u)\) to \(\cos(4u)\).

In our example, simplification is done by reworking the original expression using trigonometric identities and properties, leading to a more manageable form, \(\frac{5}{2} [\cos(6u) + \cos(4u)]\). Simplifying expressions is a key skill in enhancing ease of calculation and understanding in mathematics.