Problem 62 - Algebra and Trigonometry

Question

Use an addition or subtraction formula to simplify the equation. Then find all solutions in the interval \([0,2 \pi) .\) $$\cos x \cos 2 x+\sin x \sin 2 x=\frac{1}{2}$$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{\pi}{3} \) and \( x = \frac{5\pi}{3} \).

Step 1: Identify Relevant Formula

The given equation is \( \cos x \cos 2x + \sin x \sin 2x = \frac{1}{2} \). Notice it follows the form of a sum-to-product identity: \( \cos A \cos B + \sin A \sin B = \cos(A - B) \). Apply this identity with \( A = x \) and \( B = 2x \), which simplifies to \( \cos(x - 2x) = \cos(-x) \).

Step 2: Simplify the Equation

Using the identity from Step 1, the expression \( \cos x \cos 2x + \sin x \sin 2x \) simplifies to \( \cos(-x) \). Since \( \cos(-x) = \cos x \), the equation simplifies to \( \cos x = \frac{1}{2} \).

Step 3: Solve for x in [0, 2π)

The equation \( \cos x = \frac{1}{2} \) has solutions in the interval \([0, 2\pi)\) at \( x = \frac{\pi}{3} \) and \( x = \frac{5\pi}{3} \) because the cosine function is positive in the first and fourth quadrants.

Key Concepts

Addition and Subtraction Formulas

Trigonometry often requires simplifying complex expressions, and one of the most useful tools for this is the addition and subtraction formulas. These formulas are vital as they allow us to express trigonometric functions of compound angles in terms of functions of individual angles.
For instance, when faced with expressions like \( \cos(A \pm B) \), these can be broken down into simpler components:

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

These identities help convert complex trigonometric equations into simpler forms, which are often easier to deal with. In our original problem, recognizing that the left side of the equation matches the form \( \cos(A - B) \) made it simpler to deduce that it could be rewritten more directly as \( \cos(-x) \). This transformation is crucial for ongoing calculations.

Sum-to-Product Identities

Sum-to-product identities are incredibly useful in transforming sums or differences into products of trigonometric functions. These identities are derived from the addition and subtraction formulas and often simplify the process by reducing the complexity of the expressions.

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

These identities effectively change a sum or difference into a product, which can further simplify an equation. In this exercise, we saw the combination \( \cos x \cos 2x + \sin x \sin 2x \), which is simplified directly using the identity \( \cos(A-B) \), leading to \( \cos(-x) \). This operation simplified the equation to a more manageable form, \( \cos x = \frac{1}{2} \), allowing us to find the solutions easily.

Solving Trigonometric Equations

The process of solving trigonometric equations often involves utilizing the identities and simplifications we have discussed.
The goal is to reduce the equation to a basic form where the solutions can be identified either from known angles on the unit circle or by further algebraic manipulation. In our problem, after using the sum-to-product identity and simplification, we arrived at \( \cos x = \frac{1}{2} \).
Solving this specific equation involves finding angles \( x \) whose cosine value is \( \frac{1}{2} \).
For the interval \([0, 2\pi)\), these solutions are:

• \( x = \frac{\pi}{3} \)
• \( x = \frac{5\pi}{3} \)

These angles are found directly by considering the properties of the cosine function, which is positive in the first and fourth quadrants. Recognizing these properties lets us identify the correct angles and therefore the solutions to the equation.