Question

The equation \(\cos \theta=\frac{1}{2}, 0 \leq \theta < 2 \pi,\) has solutions \(\frac{\pi}{3}\) and \(\frac{5 \pi}{3} .\) Suppose the domain is not restricted. a) What is the general solution corresponding to \(\theta=\frac{\pi}{3} ?\) b) What is the general solution corresponding to \(\theta=\frac{5 \pi}{3} ?\)

Short answer

The general solutions are \( \theta = \frac{\pi}{3} + 2\pi n \) and \( \theta = \frac{5 \pi}{3} + 2\pi n \).

Step-by-step solution

Understanding the problem

We need to find the general solutions for \( \theta \) given the equation \( \cos \theta = \frac{1}{2} \) for all \( \theta \).

Identify known solutions in one period

The given solutions within the interval \(0 \leq \theta < 2\pi\) are \( \frac{\pi}{3} \) and \( \frac{5 \pi}{3} \).

Determine the period of cosine function

The cosine function has a period of \(2\pi\). Therefore, any solution repeats every \(2\pi\) units.

Find the general solution for \( \theta = \frac{\pi}{3} \)

Since the cosine function is periodic, the general solution is given by \( \theta = \frac{\pi}{3} + 2\pi n \), where \( n \) is any integer.

Find the general solution for \( \theta = \frac{5 \pi}{3} \)

Similarly, the general solution corresponding to \( \theta = \frac{5 \pi}{3} \) is given by \( \theta = \frac{5 \pi}{3} + 2\pi n \), where \( n \) is any integer.

Key concepts

cosine function

The cosine function, often written as \(\text{cos} \theta\), is one of the basic trigonometric functions. It's defined as the x-coordinate of a point on the unit circle. The unit circle is a circle with radius 1 centered at the origin of a coordinate plane.

Here are a few important features of the cosine function:

• The range of the cosine function is \([-1, 1]\).
• Its graph is a wave that oscillates between 1 and -1.
• The cosine function is even, meaning \(\text{cos}(-\theta) = \text{cos} \theta\).
• It has a period of \(\text{2\pi}\), which means the function repeats every \(\text{2\pi}\) units.

periodic functions

A periodic function repeats its values at regular intervals. The time it takes for a function to repeat is called its period. The sine and cosine functions are well-known examples of periodic functions.

For the cosine function, the period is \(\text{2\pi}\). This means that if you know the value of \(\text{cos} \theta\) at any given \(\theta\), you also know the values of \(\text{cos} \theta + \text{2\pi}n\) for any integer \(\text{n}\). Simply put, the graph of \(\text{cos} \theta\) looks the same between \(\text{0 to 2\pi}\), \(\text{2\pi to 4\pi}\), and so on.

This concept is critical when solving trigonometric equations because it allows us to find all possible solutions by adding multiples of the period.

general solutions

The general solution of a trigonometric equation helps us find all possible solutions, not just the ones in a restricted interval. Let's take an example with the equation \(\text{cos} \theta = \frac{1}{2}\) and \( \theta \) in the interval \( \text{0 ≤ \theta < 2\pi}\). In this case, the solutions are \( \frac{ \text{\pi} }{3}\) and \( \frac{ \text{5\pi} }{3}\).

When the domain is not restricted:

• The general solution for \( \theta = \frac{ \text{\pi} }{3}\) is \( \theta = \frac{ \text{\pi} }{3} + 2 \text{\pi} n \, \text{where n is any integer.} \)
  
• Similarly, the general solution for \( \theta = \frac{ \text{5\pi} }{3}\) is \( \theta = \frac{ \text{5\pi} }{3} + 2 \text{\pi} n \, \text{where n is any integer.} \)

Understanding the general solution allows us to see how the values repeat because of the periodic nature of trigonometric functions.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. These identities are crucial tools in solving trigonometric equations.

Some basic trigonometric identities include:

• \( \text{cos}^2 \theta + \text{sin}^2 \theta = 1 \)
• \( \text{1 + tan}^2 \theta = \text{sec}^2 \theta \)
• \( \text{cos}( \text{\theta \pm \phi} ) = \text{cos} \theta \text{cos} \phi \mp \text{sin} \theta \text{sin} \phi \)

These identities allow us to rewrite complex expressions in simpler forms.
For example, if solving \( \text{cos}(\text{\theta}) = \text{{1}\backslash {2}} \), we use the known values from the unit circle, helping us find solutions in different intervals.