Solving 2cos(θ) - 1 = 0 Trigonometric Equation In The Interval 0 ≤ Θ < 2π

Introduction

In the realm of trigonometry, solving equations is a fundamental skill. Trigonometric equations often involve finding angles that satisfy a given relationship between trigonometric functions. This article delves into the process of solving the trigonometric equation 2cos(θ) - 1 = 0 within the interval 0 ≤ θ < 2π. This interval represents one full rotation around the unit circle, encompassing all possible solutions within a single period. Trigonometric equations are crucial in various fields, including physics, engineering, and computer graphics, where periodic phenomena are modeled. Understanding how to find solutions within a specified interval is essential for practical applications. The unit circle, a circle with a radius of 1 centered at the origin, plays a central role in visualizing and solving trigonometric equations. Points on the unit circle correspond to angles and their associated cosine and sine values. The cosine of an angle is represented by the x-coordinate of the point where the terminal side of the angle intersects the unit circle, while the sine is represented by the y-coordinate. By understanding the relationship between angles and coordinates on the unit circle, we can effectively solve trigonometric equations.

Understanding the Basics of Trigonometric Equations

To solve trigonometric equations effectively, we first need to understand the fundamental trigonometric functions: sine, cosine, and tangent. The sine function, denoted as sin(θ), represents the ratio of the opposite side to the hypotenuse in a right-angled triangle. The cosine function, denoted as cos(θ), represents the ratio of the adjacent side to the hypotenuse. The tangent function, denoted as tan(θ), represents the ratio of the opposite side to the adjacent side. These functions are periodic, meaning their values repeat at regular intervals. This periodicity is crucial when solving trigonometric equations, as it implies that there can be multiple solutions within a given range. The periodicity of trigonometric functions is a key characteristic that distinguishes them from algebraic functions. For example, the sine and cosine functions have a period of 2π, which means their values repeat every 2π radians. The tangent function has a period of π, meaning its values repeat every π radians. This periodic nature leads to multiple solutions for trigonometric equations within a given interval, making it essential to consider all possible angles that satisfy the equation. The unit circle provides a visual aid for understanding the periodic behavior of trigonometric functions. As an angle θ increases, the point on the unit circle corresponding to that angle traces a circular path. The sine and cosine values of θ change as the point moves around the circle, repeating their values after each full rotation (2π radians). This visual representation helps in identifying multiple solutions and understanding the symmetry inherent in trigonometric functions.

Solving the Equation 2cos(θ) - 1 = 0

Now, let's focus on solving the specific equation 2cos(θ) - 1 = 0. Our goal is to find all values of θ within the interval 0 ≤ θ < 2π that satisfy this equation. The first step in solving any equation is to isolate the variable. In this case, we need to isolate the cosine function. To isolate cos(θ), we'll start by adding 1 to both sides of the equation:

2cos(θ) - 1 + 1 = 0 + 1

This simplifies to:

2cos(θ) = 1

Next, we divide both sides by 2:

2cos(θ) / 2 = 1 / 2

Which gives us:

cos(θ) = 1/2

Now that we have isolated cos(θ), we need to find the angles θ in the interval 0 ≤ θ < 2π whose cosine is 1/2. To find the angles whose cosine is 1/2, we can refer to the unit circle or our knowledge of common trigonometric values. The cosine function corresponds to the x-coordinate on the unit circle. We are looking for points on the unit circle where the x-coordinate is 1/2. Recall the unit circle and the common angles where cosine values are known. The cosine function is positive in the first and fourth quadrants. We know that cos(π/3) = 1/2. This gives us one solution in the first quadrant. Since cosine is also positive in the fourth quadrant, we need to find the angle in the fourth quadrant that has the same reference angle. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. In this case, the reference angle is π/3. To find the angle in the fourth quadrant, we can subtract the reference angle from 2π:

2π - π/3 = (6π/3) - (π/3) = 5π/3

Therefore, cos(5π/3) = 1/2. So, the solutions to the equation 2cos(θ) - 1 = 0 in the interval 0 ≤ θ < 2π are θ = π/3 and θ = 5π/3.

Identifying Solutions within the Interval 0 ≤ θ < 2π

After finding the potential solutions, it's crucial to verify that they lie within the specified interval, which in this case is 0 ≤ θ < 2π. This interval represents one full rotation around the unit circle, starting from 0 radians and ending just before 2π radians. Any solution outside this interval is not valid for our problem. As we found in the previous section, the potential solutions for cos(θ) = 1/2 are θ = π/3 and θ = 5π/3. Both of these angles are within the interval 0 ≤ θ < 2π. π/3 is approximately 1.047 radians, which is clearly between 0 and 2π (approximately 6.283 radians). Similarly, 5π/3 is approximately 5.236 radians, which also falls within the interval. Therefore, both solutions are valid. However, it's important to note that if we had found solutions outside this interval, we would need to adjust them by adding or subtracting multiples of 2π until they fall within the desired range. For instance, if we had a solution of θ = 7π/3, we would subtract 2π to get π/3, which is within the interval. This process of adjusting solutions ensures that we only consider angles within the specified domain.

General Solutions and the Unit Circle

While we have found the solutions within the interval 0 ≤ θ < 2π, it's important to understand the concept of general solutions for trigonometric equations. Since trigonometric functions are periodic, there are infinitely many solutions to any trigonometric equation. The solutions we found earlier, θ = π/3 and θ = 5π/3, are the principal solutions within the interval 0 ≤ θ < 2π. However, we can add multiples of 2π to these solutions and still obtain valid solutions. The general solution for cos(θ) = 1/2 can be expressed as:

θ = π/3 + 2πk and θ = 5π/3 + 2πk, where k is an integer.

This means that for any integer value of k, the resulting angle θ will satisfy the equation. For example, if k = 1, we get:

θ = π/3 + 2π = 7π/3

θ = 5π/3 + 2π = 11π/3

These are also solutions to the equation, but they lie outside the interval 0 ≤ θ < 2π. The unit circle provides a visual representation of these general solutions. Each time we add 2π to an angle, we complete a full rotation around the unit circle and return to the same point. Therefore, the cosine value remains the same. By understanding general solutions, we can express all possible solutions to a trigonometric equation, taking into account the periodic nature of trigonometric functions.

Using the Unit Circle for Visualization

The unit circle is an invaluable tool for visualizing and solving trigonometric equations. As mentioned earlier, it's a circle with a radius of 1 centered at the origin in the coordinate plane. The x-coordinate of a point on the unit circle represents the cosine of the angle, and the y-coordinate represents the sine of the angle. By visualizing the equation cos(θ) = 1/2 on the unit circle, we can easily identify the angles that satisfy the equation. We are looking for points on the unit circle where the x-coordinate is 1/2. Drawing a vertical line at x = 1/2 intersects the unit circle at two points. These points correspond to the angles π/3 and 5π/3, which we found earlier. The unit circle also helps in understanding the symmetry of trigonometric functions. The cosine function is symmetric about the x-axis, which means that if cos(θ) = 1/2, then cos(-θ) = 1/2 as well. In our case, cos(π/3) = 1/2, and the angle in the fourth quadrant with the same reference angle is 5π/3, which is equivalent to -π/3 in terms of a negative angle. This symmetry simplifies the process of finding all solutions within a given interval. The unit circle is a fundamental concept in trigonometry, and mastering its use is essential for solving trigonometric equations and understanding the behavior of trigonometric functions.

Conclusion

In this article, we've walked through the process of solving the trigonometric equation 2cos(θ) - 1 = 0 within the interval 0 ≤ θ < 2π. We began by isolating the cosine function and then identified the angles whose cosine is 1/2 using our knowledge of common trigonometric values and the unit circle. We found two solutions within the specified interval: θ = π/3 and θ = 5π/3. We also discussed the concept of general solutions, which account for the periodic nature of trigonometric functions, and how to express all possible solutions to the equation. The unit circle is a powerful tool for visualizing trigonometric functions and solving equations. By understanding the relationship between angles and coordinates on the unit circle, we can easily identify solutions and understand the symmetry inherent in trigonometric functions. Solving trigonometric equations is a fundamental skill in mathematics and has applications in various fields, including physics, engineering, and computer graphics. Mastering these techniques allows us to model and understand periodic phenomena in the real world. By practicing and applying these methods, you can confidently tackle a wide range of trigonometric equations and problems.