Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Use sum-to-product formulas to find the solutions of the equation. $$ \cos 3 x=-\cos 6 x $$

Step-by-Step Solution

Answer

Solutions are \(x = \frac{2n\pi}{9}\) and \(x = -\frac{2m\pi}{3}\) for integers \(n, m\).

Step 1: Identify the formula

We know that if \(\cos A = -\cos B\), we can use the identity \(\cos A + \cos B = 0\) which transforms to \(-2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) = 0\) to find the equations to solve.

Step 2: Apply the formula

Start by applying the sum-to-product formula: \(\cos 3x + \cos 6x = 0 \rightarrow -2\sin\left(\frac{3x + 6x}{2}\right) \sin\left(\frac{3x - 6x}{2}\right) = 0\). This simplifies to \(-2\sin\left(\frac{9x}{2}\right) \sin\left(-\frac{3x}{2}\right) = 0\).

Step 3: Set each factor to zero

We set each factor of the equation to zero:1. \(\sin\left(\frac{9x}{2}\right) = 0\)2. \(\sin\left(-\frac{3x}{2}\right) = 0\)

Step 4: Solve first equation

For \(\sin\left(\frac{9x}{2}\right) = 0\), this is true when \(\frac{9x}{2} = n\pi\) for integer \(n\). Solving for \(x\), we get \(x = \frac{2n\pi}{9}\).

Step 5: Solve second equation

For \(\sin\left(-\frac{3x}{2}\right) = 0\), this is true when \(-\frac{3x}{2} = m\pi\) for integer \(m\). Solving for \(x\), we get \(x = -\frac{2m\pi}{3}\).

Step 6: Combine solutions

The general solution will be the combination of both solutions: \{ x = \frac{2n\pi}{9} \} \cup \{ x = -\frac{2m\pi}{3} \}, where \(n, m\) are integers.

Key Concepts

Trigonometric Identities

In the world of trigonometry, identities are equations that hold true for all angles. They serve as tools that simplify complex trigonometric expressions and solve equations. In the case of cosine, an important set of identities is the sum-to-product formulas. These formulas allow us to transform the sum of trigonometric functions into products. This transformation is particularly useful in solving equations where expressions like \(\cos A + \cos B = 0\) appear.

When given \(\cos A = -\cos B\), we can convert it using \(\cos A + \cos B = 0\). This is then transformed further into the equation \(-2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) = 0\). Here, if either sine term equals zero, the original equation holds true. Understanding this identity helps transform problems into simpler forms that are easier to manage.

Solving Trigonometric Equations

Solving trigonometric equations involves finding all the angles that satisfy an equation. After transforming the original equation using trigonometric identities, the next step is to solve the resulting equations.

In our exercise, the equation \(\cos 3x = -\cos 6x\) was transformed using sum-to-product formulas into \(-2 \sin\left(\frac{9x}{2}\right) \sin\left(-\frac{3x}{2}\right) = 0\). We set each factor equal to zero:

• \(\sin\left(\frac{9x}{2}\right) = 0\)
• \(\sin\left(-\frac{3x}{2}\right) = 0\)

Each of these simpler equations is solved independently by taking advantage of the property that \(\sin\theta = 0\) leads to solutions of the form \(\theta = n\pi\), where \(n\) is an integer. This logical approach breaks down a potentially daunting problem into manageable chunks.

General Solution of Equations

The general solution in trigonometric equations refers to the complete set of solutions, often involving one or more angles, that satisfy the equation for all permissible integer values. Once individual solutions are found using identities and simplifications, these solutions are often expressed with a parameter representing any integer.

From our exercise, after solving the individual equations \(\sin\left(\frac{9x}{2}\right) = 0\) and \(\sin\left(-\frac{3x}{2}\right) = 0\), the solutions were:

• \(x = \frac{2n\pi}{9}\)
• \(x = -\frac{2m\pi}{3}\)

Here, \(n\) and \(m\) are integers, meaning these solutions represent an infinite number of angle measures that satisfy the original equation. By expressing solutions in this form, we're acknowledging that trigonometric functions are periodic, and thus repeat their values for different cycles of angles. This allows us to provide a comprehensive answer to the equation.