Problem 40 - Algebra and Trigonometry

Question

\(39-42\) = Use an Addition or Subtraction Formula to simplify the equation. Then find all solutions in the interval \([0,2 \pi)\) \(\cos \theta \cos 2 \theta+\sin \theta \sin 2 \theta=\frac{1}{2}\)

Step-by-Step Solution

Answer

Solutions are \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \) in the interval \([0, 2\pi)\).

Step 1: Identify the Given Expression

We are given the trigonometric equation \( \cos \theta \cos 2\theta + \sin \theta \sin 2\theta = \frac{1}{2} \).

Step 2: Apply Addition Formula

Use the addition formula for cosine: \( \cos(A - B) = \cos A \cos B + \sin A \sin B \). Here, let \( A = 2\theta \) and \( B = \theta \). This gives us the equation as \( \cos(2\theta - \theta) = \cos \theta \cos 2\theta + \sin \theta \sin 2\theta \).

Step 3: Simplify the Equation

Simplifying the left side of the equation \( \cos(2\theta - \theta) \) becomes \( \cos(\theta) \). So the equation simplifies to \( \cos \theta = \frac{1}{2} \).

Step 4: Find General Solutions for \( \cos \theta = \frac{1}{2} \)

The general solutions for \( \cos \theta = \frac{1}{2} \) are \( \theta = \frac{\pi}{3} + 2k\pi \) and \( \theta = \frac{5\pi}{3} + 2k\pi \), where \( k \) is an integer.

Step 5: Determine Solutions in the Interval \\( [0, 2\ pi) \\)

Within the interval \( [0, 2\pi) \), the solutions are \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \).

Key Concepts

Addition Formulas

Addition formulas help simplify expressions involving trigonometric functions. They are vital tools in trigonometry that allow you to transform the sum or difference of angles into a product of trigonometric functions. For example, the cosine addition formula is expressed as:

• \( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \)

In our exercise, we use the addition formula to simplify the expression \( \cos \theta \cos 2\theta + \sin \theta \sin 2\theta \). By applying the cosine subtraction formula \( \cos(A - B) = \cos A \cos B + \sin A \sin B \), and letting \( A = 2\theta \) and \( B = \theta \), the given expression simplifies. This eventually leads to a much simpler equation: \( \cos \theta = \frac{1}{2} \).
Simplifying trigonometric expressions using addition formulas often allows us to solve equations more easily and efficiently.

Cosine Function

The cosine function, \( \cos \theta \), is one of the primary trigonometric functions. It represents the horizontal coordinate of a point on the unit circle. Crucially, it is periodic with a period of \( 2\pi \), meaning that \( \cos(\theta) = \cos(\theta + 2k\pi) \) for any integer \( k \). This property is used extensively when finding solutions to trigonometric equations.
Understanding how \( \cos \theta \) behaves within its period, \([0, 2\pi)\), helps in solving problems like finding solutions to equations like \( \cos \theta = \frac{1}{2} \). Within one complete cycle, \( \cos \theta \) will take each of its values twice, once between \( 0 \) and \( \pi \) and once between \( \pi \) and \( 2\pi \). For \( \cos \theta = \frac{1}{2} \), the solutions within this interval are at \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \). Recognizing where these values occur on the unit circle can simplify solving such equations.

Trigonometric Equations

Trigonometric equations involve trigonometric functions and require solving for the angle variable, often within a specific interval such as \([0, 2\pi)\). A key strategy in solving these equations is recognizing identities and using fundamental properties of trigonometric functions.
For the equation \( \cos \theta = \frac{1}{2} \), the goal is to find all \( \theta \) that satisfy this within the interval. Each angle in standard position has an equivalent angle formed by adding full rotations of \( 2\pi \). This requires applying the knowledge of the unit circle and the periodic nature of trigonometric functions. When we solve \( \cos \theta = \frac{1}{2} \), we recognize from the unit circle that \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \) within \( [0, 2\pi) \). Adding multiple rotations (\( 2k\pi \)) would give more general solutions, however, we restrict it to the given interval in this case.
Trigonometric equations often have multiple solutions, and understanding periods and identities is the key to finding all possible solutions satisfactorily.