Problem 18 - Algebra and Trigonometry

Question

\(17-24\) n Solve the given equation, and list six specific solutions. $$ \cos \theta=\frac{1}{2} $$

Step-by-Step Solution

Answer

Solutions: \(\frac{\pi}{3}, \frac{7\pi}{3}, -\frac{5\pi}{3}, \frac{5\pi}{3}, \frac{11\pi}{3}, -\frac{1\pi}{3}\).

Step 1: Identify the angles for cosine value

The cosine of angle \(\theta\) is equal to \(\frac{1}{2}\). We know from trigonometric knowledge that this corresponds to the angles \(\theta = \frac{\pi}{3}\) and \(\theta = \frac{5\pi}{3}\) in the range \([0, 2\pi]\).

Step 2: Determine the general solutions

To express these specific angles as a more general solution, consider the periodic nature of the cosine function, which has a period of \(2\pi\). Hence, any angle that differs by an integer multiple of \(2\pi\) will also satisfy the equation. Therefore, the general solutions are \(\theta = \frac{\pi}{3} + 2n\pi\) and \(\theta = \frac{5\pi}{3} + 2n\pi\), where \(n\) is an integer.

Step 3: List six specific solutions

By substituting different integer values for \(n\), we can obtain six specific solutions. Let's select \(n = 0, \pm 1, \pm 2\) and apply to both general solutions: For \(\theta = \frac{\pi}{3} + 2n\pi\):1. \(n = 0\): \(\theta = \frac{\pi}{3}\)2. \(n = 1\): \(\theta = \frac{7\pi}{3}\)3. \(n = -1\): \(\theta = -\frac{5\pi}{3}\)For \(\theta = \frac{5\pi}{3} + 2n\pi\):4. \(n = 0\): \(\theta = \frac{5\pi}{3}\)5. \(n = 1\): \(\theta = \frac{11\pi}{3}\)6. \(n = -1\): \(\theta = -\frac{1\pi}{3}\)

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions that deals with the ratios of the sides of a right triangle. It's often represented by the symbol \(\cos\). This function measures the adjacent side over the hypotenuse in a right triangle. In the unit circle, cosine corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle.
The cosine function is periodic and has a period of \(2\pi\), meaning that it repeats its values every \(2\pi\) units. Its range is from -1 to 1, making it ideal for problems involving oscillation or waves. When solving equations like \(\cos \theta = \frac{1}{2}\), we're essentially finding the angles \(\theta\) that provide this cosine ratio. Different quadrant rules apply, which affect the sign and value of the cosine function, playing a critical role in determining possible angle solutions.

Angle Solutions

Angle solutions refer to the process of finding angles \(\theta\) that satisfy a given trigonometric equation. For the equation \(\cos \theta = \frac{1}{2}\), we identify angles such as \(\theta = \frac{\pi}{3}\) and \(\theta = \frac{5\pi}{3}\) within the interval \([0, 2\pi]\). These are the angle solutions for one cycle around the unit circle.
Remember, the unit circle is a fundamental tool in trigonometry, providing a visual way to understand angle measures and their corresponding sine and cosine values. It's crucial to recognize that for each cosine value, there could be multiple angle solutions due to the symmetrical properties of the cosine function around the circle's center.

General Solutions

General solutions represent a method of expressing all possible solutions to a trigonometric equation by incorporating the periodic nature of trigonometric functions. Since the cosine function is periodic with a period of \(2\pi\), any angle \(\theta\) plus or minus an integer multiple of \(2\pi\) will satisfy the equation \(\cos \theta = \frac{1}{2}\).
We can write these general solutions as \(\theta = \frac{\pi}{3} + 2n\pi\) and \(\theta = \frac{5\pi}{3} + 2n\pi\), where \(n\) is any integer. This implies that by adding or subtracting these multiples, we can find all possible angles that will give the same cosine value, \(\frac{1}{2}\). General solutions are particularly useful for solving equations over different ranges or multiple cycles.

Periodic Functions

Periodic functions, like the cosine function, repeat their values in evenly spaced intervals called periods. For the cosine function, this period is \(2\pi\).
Understanding periodicity is crucial as it allows us to extend solutions over infinite intervals of angles. This is because once the function completes a full cycle, it starts repeating the same values. For example, in the cosine function, \(\cos(\theta) = \cos(\theta + 2\pi n)\) for any integer \(n\). Thus, this concept of periodicity makes it easier to solve trigonometric equations by identifying patterns and extending solutions.

Specific Solutions

Specific solutions are those which give us actual numerical or analytical values for the angles satisfying the equation for selected integers \(n\). Once we have derived the general solution, we can substitute various integral values for \(n\) to acquire specific solutions.
For our equation \(\cos \theta = \frac{1}{2}\), we determined that the general solutions were \(\theta = \frac{\pi}{3} + 2n\pi\) and \(\theta = \frac{5\pi}{3} + 2n\pi\). By inserting \(n = 0, \pm 1, \pm 2\), we populating specific solutions, like \(\frac{\pi}{3}, \frac{7\pi}{3}, \frac{5\pi}{3}\), and more. This process ensures a comprehensive set of solutions that align with the periodic nature of trigonometric functions.