Problem 14 - Algebra and Trigonometry

Question

express each sum or difference as a product. If possible, find this product’s exact value. $$ \cos 9 x-\cos 7 x $$

Step-by-Step Solution

Answer

The given expression \(\cos 9x - \cos 7x\) can be expressed as a product \(2 \sin (8x) \sin(x)\).

Step 1: Identify the Trigonometric Formula

The key formula to use here is \(\cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{B - A}{2}\right)\). In this case, \(A = 9x\) and \(B = 7x\).

Step 2: Substitute into the Formula

Substitute the values of A and B into the formula: \(-2 \sin\left(\frac{9x + 7x}{2}\right) \sin\left(\frac{7x - 9x}{2}\right)\)

Step 3: Simplify the Expression

Upon simplifying the expression, we get: \(-2 \sin (8x) \sin (-x)\). Since \(\sin (-x) = - \sin(x)\), the expression further simplifies to \(2 \sin (8x) \sin(x)\).

Key Concepts

Cosine Difference Formula

The cosine difference formula is a fundamental tool in trigonometry that allows the transformation of the difference between two cosine functions into a product of sine functions. Specifically, the formula states that:
\[\begin{equation} \[\cos(A) - \cos(B) = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)\]\end{equation}\]
When you encounter an expression like \(\cos(9x) - \cos(7x)\), you can apply this formula. Substitute \(A = 9x\) and \(B = 7x\) into the cosine difference formula to simplify the given expression into a product involving sines.

Understanding and applying the cosine difference formula effectively requires recognizing the form \(\cos(A) - \cos(B)\) and performing correct substitutions. Improving with practice, you'll streamline complex trigonometric expressions with ease.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is all about making complex or cumbersome formulas easier to work with and understand. You might need to apply various trigonometric identities, laws, or formulas to break down expressions into simpler components.

For instance, once you've applied the cosine difference formula to the trigonometric sum \(\cos(9x) - \cos(7x)\), you'll get \(-2 \sin(8x) \sin(-x)\). Here, it's helpful to recognize that the sine function is odd, meaning that \(\sin(-x) = -\sin(x)\). This crucial insight allows you to further simplify the expression to \(2 \sin(8x) \sin(x)\). This step not only makes the expression cleaner, but also paves the way for further simplifications or evaluations.

Simplifying trigonometric expressions often involves a series of small, careful steps that, when correctly executed, lead to a much tidier and more workable mathematical statement.

Trigonometric Identities

Trigonometric identities are the backbone of trigonometry. They are equations that are true for all values of the variables involved, providing powerful tools to solve and simplify trigonometric expressions and equations.

These identities include reciprocal identities, Pythagorean identities, even-odd identities, sum and difference formulas, double and half-angle formulas, to name a few. For example, the identity \(\sin(-x) = -\sin(x)\) used in simplifying the expression \(\cos(9x) - \cos(7x)\) falls under the category of even-odd identities.

Memorizing these identities can significantly help in solving trigonometric problems. However, understanding how and when to apply them is even more crucial. This involves practice and often recognizing patterns in trigonometric expressions that match the form of known identities. With enough experience, you can manipulate these expressions quickly to arrive at the simplest form or to solve equations efficiently.