Solving 2cos Θ + 1 = 0 Find Solutions In [0, 2π)

Introduction

In the realm of trigonometry, solving equations is a fundamental skill. Trigonometric equations, which involve trigonometric functions such as sine, cosine, and tangent, often arise in various mathematical and scientific contexts. Solving trigonometric equations requires a solid understanding of trigonometric identities, unit circles, and the periodic nature of trigonometric functions. In this comprehensive guide, we will delve into the process of solving the trigonometric equation $2 \cos \theta + 1 = 0$ within the interval $[0, 2\pi)$. This interval represents one complete revolution around the unit circle, and our goal is to identify all angles $\theta$ within this range that satisfy the given equation. By systematically applying trigonometric principles and algebraic techniques, we will determine the solutions and express them in radians in terms of $\pi$, adhering to the requested format. This exploration will not only enhance your problem-solving skills but also deepen your understanding of the behavior of trigonometric functions. Mastering the art of solving trigonometric equations is crucial for anyone pursuing further studies in mathematics, physics, engineering, or any field where periodic phenomena are encountered.

Understanding the Problem

The given equation is $2 \cos \theta + 1 = 0$, and we are tasked with finding all solutions for $\theta$ within the interval $[0, 2\pi)$. This means we are looking for angles between 0 radians and $2\pi$ radians (which is equivalent to 360 degrees) that make the equation true. The equation involves the cosine function, which represents the x-coordinate of a point on the unit circle. Understanding the unit circle is paramount in solving trigonometric equations, as it provides a visual representation of the cosine, sine, and tangent values for different angles. The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the Cartesian coordinate system. The angle $\theta$ is measured counterclockwise from the positive x-axis. The cosine of $\theta$, denoted as $\cos \theta$, is the x-coordinate of the point where the terminal side of the angle intersects the unit circle. To solve the equation, we need to isolate the cosine function and determine the angles at which the cosine value is equal to the resulting value. This involves algebraic manipulation and a thorough understanding of the cosine function's behavior across different quadrants of the unit circle. Furthermore, recognizing the periodicity of the cosine function is crucial, as it repeats its values every $2\pi$ radians. However, since we are only interested in solutions within the interval $[0, 2\pi)$, we need to identify the specific angles within this range that satisfy the equation. By systematically analyzing the equation and leveraging our knowledge of the unit circle, we can effectively find all possible solutions.

Solving the Equation

To solve the equation $2 \cos \theta + 1 = 0$, we first need to isolate the cosine function. This involves a series of algebraic steps that will lead us to a simplified form where we can directly determine the values of $\theta$. The initial equation is $2 \cos \theta + 1 = 0$. Our first step is to subtract 1 from both sides of the equation: $2 \cos \theta = -1$. Next, we divide both sides by 2 to isolate the cosine function: $\cos \theta = -\frac{1}{2}$. Now, we have the simplified equation $\cos \theta = -\frac{1}{2}$. This equation tells us that we are looking for angles $\theta$ where the cosine value is equal to $-1/2$. To find these angles, we need to consider the unit circle. The cosine function is negative in the second and third quadrants of the unit circle. We know that $\cos(\frac{\pi}{3}) = \frac{1}{2}$, which is a reference angle. The angles in the second and third quadrants that have a reference angle of $\frac{\pi}{3}$ are the solutions to our equation. In the second quadrant, the angle is $\pi - \frac{\pi}{3} = \frac{2\pi}{3}$. In the third quadrant, the angle is $\pi + \frac{\pi}{3} = \frac{4\pi}{3}$. Therefore, the solutions to the equation $2 \cos \theta + 1 = 0$ in the interval $[0, 2\pi)$ are $\theta = \frac{2\pi}{3}$ and $\theta = \frac{4\pi}{3}$. These angles correspond to the points on the unit circle where the x-coordinate is $-1/2$.

Finding Solutions within the Interval [0, 2π)

After isolating the cosine function and determining that $\cos \theta = -\frac{1}{2}$, the next crucial step is to identify the angles $\theta$ within the specified interval of $[0, 2\pi)$ that satisfy this condition. This interval represents one complete revolution around the unit circle, starting from 0 radians and ending at $2\pi$ radians. To find the solutions, we need to consider the quadrants of the unit circle where the cosine function is negative, as $-1/2$ is a negative value. The cosine function is negative in the second and third quadrants. In the second quadrant, angles lie between $\frac{\pi}{2}$ and $\pi$, while in the third quadrant, angles lie between $\pi$ and $\frac{3\pi}{2}$. We know that the reference angle for $\cos^{-1}(\frac{1}{2})$ is $\frac{\pi}{3}$. This means that the angles we are looking for have a reference angle of $\frac{\pi}{3}$ in the second and third quadrants. To find the angle in the second quadrant, we subtract the reference angle from $\pi$: $\theta_1 = \pi - \frac{\pi}{3} = \frac{3\pi}{3} - \frac{\pi}{3} = \frac{2\pi}{3}$. To find the angle in the third quadrant, we add the reference angle to $\pi$: $\theta_2 = \pi + \frac{\pi}{3} = \frac{3\pi}{3} + \frac{\pi}{3} = \frac{4\pi}{3}$. Therefore, the solutions within the interval $[0, 2\pi)$ are $\theta = \frac{2\pi}{3}$ and $\theta = \frac{4\pi}{3}$. These angles are the only angles within the given interval where the cosine function has a value of $-1/2$.

Expressing the Solutions in Radians

Having identified the solutions as $\theta = \frac{2\pi}{3}$ and $\theta = \frac{4\pi}{3}$, it is essential to express these solutions correctly in radians, as requested by the problem statement. Radians are a unit of angular measure that relate the arc length of a circle to its radius. One radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. In the context of the unit circle, which has a radius of 1, the radian measure of an angle is simply the length of the arc it subtends. Expressing angles in radians is crucial in many areas of mathematics and physics, particularly when dealing with trigonometric functions, calculus, and rotational motion. The conversion between degrees and radians is given by the relationship $2\pi \text{ radians} = 360 \text{ degrees}$, or equivalently, $\pi \text{ radians} = 180 \text{ degrees}$. Our solutions, $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$, are already expressed in radians and are in terms of $\pi$, which is the desired format. The angle $\frac{2\pi}{3}$ radians corresponds to 120 degrees, and the angle $\frac{4\pi}{3}$ radians corresponds to 240 degrees. Both of these angles lie within the interval $[0, 2\pi)$, which is equivalent to $[0, 360)$ degrees. Therefore, we can confidently state that the solutions to the equation $2 \cos \theta + 1 = 0$ in the interval $[0, 2\pi)$, expressed in radians in terms of $\pi$, are $\theta = \frac{2\pi}{3}$ and $\theta = \frac{4\pi}{3}$.

Final Answer

In conclusion, to find all solutions of the equation $2 \cos \theta + 1 = 0$ in the interval $[0, 2\pi)$, we followed a systematic approach involving algebraic manipulation, understanding the unit circle, and applying trigonometric principles. We began by isolating the cosine function, which led us to the equation $\cos \theta = -\frac{1}{2}$. We then recognized that the cosine function is negative in the second and third quadrants of the unit circle. By considering the reference angle and the properties of the unit circle, we identified the two solutions within the given interval: $\theta = \frac{2\pi}{3}$ and $\theta = \frac{4\pi}{3}$. These solutions were already expressed in radians in terms of $\pi$, which was the required format. The final answer, therefore, is $\theta = \frac{2\pi}{3}, \frac{4\pi}{3}$. This result demonstrates the importance of a strong foundation in trigonometry and algebraic techniques for solving trigonometric equations. It also highlights the significance of the unit circle as a visual aid in understanding the behavior of trigonometric functions. By mastering these concepts, one can confidently tackle a wide range of trigonometric problems and applications.

$\theta=\frac{2\pi}{3}, \frac{4\pi}{3}$