Problem 2 - Precalculus with Limits

Question

The equation \( 2 \sin \theta + 1 = 0 \) has the solutions \( \theta = \frac{7\pi}{6} + 2n\pi \) and \( \theta = \frac{11\pi}{6} + 2n\pi \), which are called ________ solutions.

Step-by-Step Solution

Answer

The solutions are called General Solutions.

Step 1: Analyzing the provided solutions

The solutions that are provided are given as \( \frac{7\pi}{6} + 2n\pi \) and \( \frac{11\pi}{6} + 2n\pi \). Notice that these solutions are in the format of a constant \( A \) added to an integer multiple of \( 2\pi \).

Step 2: Recognizing the type of solutions

Solutions that are given in the format of \( A + 2n\pi \) where \( A \) is a constant and \( n \) is an integer are called general solutions in trigonometry. Each particular integer value of \( n \) specifies a particular solution.

Key Concepts

Trigonometric Equations

Trigonometric equations are mathematical expressions that involve trigonometric functions such as sine, cosine, or tangent. These equations often appear in problems related to angles and periodic phenomena. In the example from the exercise, we have a trigonometric equation, \( 2 \sin \theta + 1 = 0 \), where the sine function relates an angle \( \theta \) to a numerical result. Solving these equations is crucial in various fields, including physics, engineering, and astronomy, as it helps in finding the angles that satisfy certain conditions.
This specific equation represents a condition where the sine of an angle, multiplied by 2 and increased by 1, results in zero. To isolate \( \sin \theta \), one would solve \( \sin \theta = -\frac{1}{2} \). Understanding what the equation is asking helps us move to the next steps, where we determine how periodicity influences the solutions.

Periodicity in Trigonometry

Periodicity is a fundamental property of trigonometric functions. It means that these functions repeat their values in regular intervals. The basic period for \( \sin \theta \) and \( \cos \theta \) is \( 2\pi \), which means these functions complete a cycle every \( 2\pi \) radians. This is important because it influences how we find multiple solutions to trigonometric equations. When solving, we accounts for this periodicity by adding \( 2n\pi \) (where \( n \) is an integer) to the solution.
In our example, the solutions \( \frac{7\pi}{6} \) and \( \frac{11\pi}{6} \) are found within one period, but due to periodicity, they can be extended by any integer multiple of \( 2\pi \) to produce an infinite number of solutions, leading to general solutions for \( \theta \). Recognizing and applying periodicity is crucial for accurately interpreting all possible repetitions of solutions.

Solving Trigonometric Equations

Solving a trigonometric equation involves several steps and an understanding of the basic characteristics of trigonometric functions. We start by isolating the trigonometric function on one side of the equation. For example, from \( 2 \sin \theta + 1 = 0 \), we simplify it to \( \sin \theta = -\frac{1}{2} \).
After isolating the function, use known values or identities to find the general solutions. For \( \sin \theta = -\frac{1}{2} \), the specific angles \( \theta \) are the arcs which solve this condition within the principal range. These principal solutions are \( \frac{7\pi}{6} \) and \( \frac{11\pi}{6} \). To find all solutions, utilize periodicity by adding \( 2n\pi \) to each principal solution: \( \theta = \frac{7\pi}{6} + 2n\pi \) and \( \theta = \frac{11\pi}{6} + 2n\pi \).
This approach considers the cyclical nature of trigonometric functions and helps develop a comprehensive set of solutions applicable to a wide range of problems, demonstrating how understanding periodicity and function properties are keys to solving such equations efficiently.