Problem 49 - Algebra and Trigonometry

Question

\(43-52\) a Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi) .\) \(\cos 2 \theta-\cos 4 \theta=0\)

Step-by-Step Solution

Answer

The solutions are \( \theta = 0, \frac{\pi}{3}, \frac{2\pi}{3} \).

Step 1: Apply Double Angle Formula

To solve the equation \( \cos 2\theta - \cos 4\theta = 0 \), we first use the double-angle formula for cosine: \( \cos 4\theta = 2\cos^2 2\theta - 1 \). Substitute this into the equation: \( \cos 2\theta - (2\cos^2 2\theta - 1) = 0 \).

Step 2: Simplify the Equation

Simplifying the expression from Step 1, we find: \( \cos 2\theta - 2\cos^2 2\theta + 1 = 0 \). Rearrange the terms to form a quadratic equation in terms of \( \cos 2\theta \): \( 2\cos^2 2\theta - \cos 2\theta - 1 = 0 \).

Step 3: Solve the Quadratic Equation

Factor the quadratic equation: \( (2\cos 2\theta + 1)(\cos 2\theta - 1) = 0 \). From the factor \( 2\cos 2\theta + 1 = 0 \), we get \( \cos 2\theta = -\frac{1}{2} \). From the factor \( \cos 2\theta - 1 = 0 \), we get \( \cos 2\theta = 1 \).

Step 4: Find Solutions in the Original Interval

For \( \cos 2\theta = -\frac{1}{2} \), the solutions in the interval \([0, 2\pi)\) are \( 2\theta = \frac{2\pi}{3} \) and \( 2\theta = \frac{4\pi}{3} \). For \( \cos 2\theta = 1 \), we have \( 2\theta = 0 \).

Step 5: Solve for \( \theta \)

Convert the solutions of \( 2\theta \) to \( \theta \): from \( 2\theta = \frac{2\pi}{3} \), we get \( \theta = \frac{\pi}{3} \); from \( 2\theta = \frac{4\pi}{3} \), we get \( \theta = \frac{2\pi}{3} \); from \( 2\theta = 0 \), we get \( \theta = 0 \).

Step 6: Final Answer

The solutions to the equation \( \cos 2\theta - \cos 4\theta = 0 \) in the interval \([0, 2\pi)\) are: \( \theta = 0, \frac{\pi}{3}, \frac{2\pi}{3} \).

Key Concepts

Double Angle Formulas

Double angle formulas are essential tools in trigonometry, particularly useful when working with expressions involving angles that are multiples of two. The double angle formula for cosine is expressed as:

• \( \cos 2\theta = 2\cos^2 \theta - 1 \)
• Alternatively, \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)

These formulas help simplify trigonometric expressions by reducing the power of trigonometric functions, like changing \( \cos 4\theta \) into an equation that depends on \( \cos 2\theta \).
In the given problem, the double angle formula \( \cos 4\theta = 2\cos^2 2\theta - 1 \) is applied to transform the expression \( \cos 4\theta \) into something more manageable. This conversion allows the equation to be simplified and eventually solved by forming a quadratic equation in terms of \( \cos 2\theta \). By leveraging double angle formulas, we pave the way to finding solutions to trigonometric equations by breaking down complex expressions into more straightforward components.

Half Angle Formulas

Half angle formulas are another set of trigonometric identities that express trigonometric functions of half angles in terms of the full angle. While they are not directly used in the solution to this exercise, understanding them is crucial when solving more complex problems. These formulas express the half angle in terms of square roots, which can be particularly useful for integrating or differentiating trigonometric functions.

• \( \cos \frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos \theta}{2}} \)
• \( \sin \frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos \theta}{2}} \)

Half angle formulas allow for flexibility in solving and manipulating trigonometric identities, making them powerful tools in calculus and trigonometry. While solving equations like \( \cos 2\theta = -\frac{1}{2} \) could be tackled using these formulas in reverse, this solution aims at breaking down and simplifying using double angle approaches specifically. Nonetheless, it's beneficial to grasp half angle concepts as they complement the double angle understanding and broaden your problem-solving toolkit.

Cosine Function

The cosine function is one of the primary trigonometric functions, fundamental in understanding angles and their properties within right-angled triangles and the unit circle. The function is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. In the context of the unit circle, it represents the x-coordinate of a point on the circle's circumference.

• Its values range from -1 to 1.
• The function is even, meaning \( \cos(-x) = \cos(x) \).
• Cosine has a period of \(2\pi\), meaning it repeats every \(2\pi\) radians.

In trigonometric equations, the cosine function assumes values over specified intervals. In the exercise, values of the cosine function such as \( \cos 2\theta = -\frac{1}{2} \) and \( \cos 2\theta = 1 \) are pivotal for finding possible solutions within a desired range. Both the unit circle and triangle relationships contribute to solving these equations by locating specific angle measures multiplies within the set interval. Recognizing these periodic shifts and symmetric properties helps when solving equations and converting between angle forms.