Problem 14 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a sum or difference. $$ \cos \theta-\cos 5 \theta $$

Step-by-Step Solution

Answer

\( \cos \theta - \cos 5\theta = 2 \sin(3\theta) \sin(2\theta) \).

Step 1: Identify the Expression

The given expression is \( \cos \theta - \cos 5\theta \). We need to express this as a sum or difference using trigonometric identities.

Step 2: Apply Product-to-Sum Identity

The product-to-sum identities provide a way to express the difference of cosines as a product. The identity is: \( \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \).

Step 3: Substitute Values into Identity

In the expression \( \cos \theta - \cos 5\theta \), let \( A = 5\theta \) and \( B = \theta \). Substitute into the identity: \[ \cos \theta - \cos 5\theta = -2 \sin\left(\frac{\theta + 5\theta}{2}\right) \sin\left(\frac{\theta - 5\theta}{2}\right) \].

Step 4: Simplify Angles

Simplify the fractions inside the sine functions: \( \frac{\theta + 5\theta}{2} = \frac{6\theta}{2} = 3\theta \) and \( \frac{\theta - 5\theta}{2} = \frac{-4\theta}{2} = -2\theta \).

Step 5: Write Final Expression

Substitute the simplified angles back into the expression: \[ \cos \theta - \cos 5\theta = -2 \sin(3\theta) \sin(-2\theta) \]. Since \( \sin(-x) = -\sin(x) \), we have \( \sin(-2\theta) = -\sin(2\theta) \). Thus, \[ \cos \theta - \cos 5\theta = 2 \sin(3\theta) \sin(2\theta) \].

Key Concepts

Product-to-Sum Identities

Trigonometric identities are truly fascinating because they can transform complex expressions into simpler forms that are easier to comprehend. The product-to-sum identities are a set of formulas that do just that by converting products of trigonometric functions into sums or differences. This process is helpful when simplifying the expression of trigonometric functions.
For example, the difference of cosines can be expressed using the identity:

• \( \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \)

This identity allows us to break down something like \( \cos \theta - \cos 5\theta \) into a product of sines. The functions of sine that appear, \( \sin\left(\frac{A+B}{2}\right) \) and \( \sin\left(\frac{A-B}{2}\right) \), offer a clearer view into the behavior of the original expression.
By understanding and applying these trigonometric identities, not only do calculations become simpler, but they also pave the way for solving more complex trigonometric equations.

Difference of Cosines

Understanding how the difference of cosines works can be a game-changer in simplifying trigonometric expressions. The formula mentioned in the previous section is specifically for managing such differences: \( \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \).
Let's break it down:

• The values \( A \) and \( B \) represent the angles involved in your cosine expressions. For instance, in \( \cos \theta - \cos 5\theta \), \( A = 5\theta \) and \( B = \theta \).
• Once identified, you plug these into the formula to find the equivalent expression.

This transformation aids in seeing the expression as a product of sines, which is often more practical for further trigonometric operations, like integration or solving equations. Moreover, this approach converts subtraction into a multiplication format, which can flow better into the algebraic manipulation.

Trigonometric Simplification

Simplifying trigonometric expressions is essential in making complex mathematics more approachable. It's like cleaning up a messy equation to see its beautiful underlying structure. By using identities like the differences of cosine, we can turn a subtraction into a multiplication, making calculations often easier to perform and analyze.
Take the expression \( \cos \theta - \cos 5\theta \). After applying the product-to-sum identity, we simplify the angles:

• \( \frac{\theta + 5\theta}{2} = 3\theta \)
• \( \frac{\theta - 5\theta}{2} = -2\theta \)

The key here is maintaining ease of computation. The more you simplify, the more you can appreciate the elegance of trigonometric forms. Keeping track of these transformations clarifies the broader picture and prepares you for more advanced operations, like integration and differentiation.

Angle Substitution

The technique of angle substitution is crucial to simplifying trigonometric expressions, especially when using identities. Substitution involves replacing angles in formulas with the given angles in your specific problem. It shifts the abstract identity into a practical form.
For example, when handling \( \cos \theta - \cos 5\theta \), you substitute \( \theta \) and \( 5\theta \) into the product-to-sum identity, thus allowing you to blend the abstract formula with the concrete problem at hand:

• Substitute \( A = 5\theta \) and \( B = \theta \) into the product-to-sum identity.
• Evaluate the simplified angle positions: \( 3\theta \) and \(-2\theta \).

It’s like fitting a specific key into a lock – once the angles match the substitution, the entire expression unlocks a new simplified form. Being comfortable with substitution and simplification takes practice but becomes an invaluable tool in handling a wide range of mathematical problems.