Solving Cos(θ) = -1/2: A Step-by-Step Guide

Introduction

In this comprehensive guide, we will delve into the process of solving the trigonometric function cos(θ) = -1/2 over the interval [0°, 360°). Trigonometric equations like this one appear frequently in various fields, including mathematics, physics, and engineering. Understanding how to solve them is a fundamental skill. Our goal here is to provide a clear, step-by-step explanation that will help you grasp the underlying concepts and confidently solve similar problems. Whether you're a student tackling trigonometry for the first time or someone looking to refresh your knowledge, this guide will offer valuable insights and practical techniques.

We'll start by understanding the basics of cosine and the unit circle, then move on to finding the reference angle and determining the quadrants where the cosine function is negative. Finally, we'll identify the solutions within the given interval and express them in degrees. By the end of this guide, you'll have a solid understanding of how to solve trigonometric equations of this type.

Understanding the Cosine Function

The cosine function, often abbreviated as cos(θ), is a fundamental concept in trigonometry. It represents the x-coordinate of a point on the unit circle corresponding to an angle θ. The unit circle is a circle with a radius of 1 centered at the origin (0, 0) in the Cartesian coordinate system. Imagine a point moving along the circumference of this circle, starting from the positive x-axis (0°). As the point moves counterclockwise, it traces an angle θ, and the cosine of that angle is simply the x-coordinate of the point's location.

The cosine function has several key characteristics. It oscillates between -1 and 1, meaning that the value of cos(θ) will always fall within this range. When θ = 0°, the point is on the positive x-axis, and cos(0°) = 1. As θ increases, cos(θ) decreases until θ = 90°, where the point is on the positive y-axis, and cos(90°) = 0. The function continues to decrease until θ = 180°, where the point is on the negative x-axis, and cos(180°) = -1. After this, cos(θ) increases again, reaching 0 at θ = 270° and returning to 1 at θ = 360°. This cyclical behavior is crucial for understanding how to solve trigonometric equations over a given interval.

To solve the equation cos(θ) = -1/2, we need to find the angles θ within the interval [0°, 360°) where the x-coordinate on the unit circle is -1/2. This involves identifying the quadrants where cosine is negative and using the reference angle to find the specific solutions. The unit circle and the properties of the cosine function are the foundational tools we'll use throughout this process. Understanding these basics will make the rest of the solution much clearer and more intuitive.

Finding the Reference Angle

The reference angle is a crucial concept in trigonometry, particularly when solving trigonometric equations. It is the acute angle formed between the terminal side of the given angle and the x-axis. In simpler terms, it’s the smallest angle that the given angle makes with the x-axis. The reference angle is always between 0° and 90°, and it helps us to determine the trigonometric values of angles in any quadrant based on their values in the first quadrant.

To find the reference angle for cos(θ) = -1/2, we first ignore the negative sign and consider cos(θ) = 1/2. We need to find the angle whose cosine is 1/2. From our knowledge of special trigonometric values, we know that cos(60°) = 1/2. Therefore, the reference angle, which we'll denote as θ_ref, is 60°. This reference angle is the cornerstone for finding the solutions in the quadrants where cosine is negative.

The significance of the reference angle lies in its ability to simplify the process of finding angles in different quadrants. By understanding the reference angle, we can determine the angles in the other quadrants that have the same trigonometric value (but possibly with a different sign). In the case of cosine, we know that the cosine function is positive in the first and fourth quadrants and negative in the second and third quadrants. This knowledge, combined with the reference angle, allows us to pinpoint the specific solutions to our equation.

The next step is to identify the quadrants where cos(θ) is negative, as this will guide us in finding the angles within the interval [0°, 360°) that satisfy the equation cos(θ) = -1/2. The reference angle of 60° will be our key to unlocking these solutions, as we'll use it to find the angles in the appropriate quadrants that have a cosine value of -1/2. Understanding how to find and use the reference angle is a vital skill in solving trigonometric problems.

Determining the Quadrants

To solve trigonometric equations effectively, it's essential to understand the signs of trigonometric functions in different quadrants. The Cartesian plane is divided into four quadrants, numbered I through IV, moving counterclockwise starting from the upper right. Each quadrant has its own unique combination of positive and negative values for sine, cosine, and tangent.

In the first quadrant (0° to 90°), all trigonometric functions (sine, cosine, and tangent) are positive. This is because both the x-coordinate and y-coordinate are positive in this quadrant. In the second quadrant (90° to 180°), sine is positive, while cosine and tangent are negative. Here, the y-coordinate is positive, but the x-coordinate is negative, making cosine negative. The third quadrant (180° to 270°) is where both sine and cosine are negative, while tangent is positive. In this quadrant, both the x-coordinate and y-coordinate are negative. Finally, in the fourth quadrant (270° to 360°), cosine is positive, while sine and tangent are negative. The x-coordinate is positive in this quadrant, while the y-coordinate is negative.

For our equation, cos(θ) = -1/2, we are looking for angles where the cosine function is negative. As we established, cosine is negative in the second and third quadrants. This means that the solutions to our equation will lie in these two quadrants. Knowing this helps us narrow down the possibilities and focus our attention on the relevant portions of the unit circle.

With the reference angle of 60° in hand, we now need to find the angles in the second and third quadrants that have a reference angle of 60°. This will give us the specific solutions to cos(θ) = -1/2 within the interval [0°, 360°). Understanding the quadrant signs of trigonometric functions is a fundamental step in solving trigonometric problems, and it helps us to accurately identify the angles that satisfy a given equation.

Identifying the Solutions within [0°, 360°)

Now that we know the reference angle is 60° and cosine is negative in the second and third quadrants, we can find the solutions to cos(θ) = -1/2 within the interval [0°, 360°). To do this, we'll use the reference angle to determine the angles in the second and third quadrants that have the same cosine value (but with a negative sign).

In the second quadrant, angles are measured from 0° to 180°. To find the angle with a reference angle of 60°, we subtract the reference angle from 180°. So, θ = 180° - 60° = 120°. Thus, one solution to the equation is 120°. This angle lies in the second quadrant, and its cosine value is indeed -1/2.

In the third quadrant, angles are measured from 180° to 270°. To find the angle with a reference angle of 60°, we add the reference angle to 180°. So, θ = 180° + 60° = 240°. This gives us another solution to the equation: 240°. This angle lies in the third quadrant, where cosine is negative, and its cosine value is -1/2.

Therefore, the solutions to the trigonometric equation cos(θ) = -1/2 within the interval [0°, 360°) are 120° and 240°. These are the only two angles within this interval where the x-coordinate on the unit circle is -1/2. It's helpful to visualize these angles on the unit circle to reinforce your understanding. You'll see that they are symmetrically placed about the y-axis, reflecting the symmetry of the cosine function.

Solving trigonometric equations often involves a combination of understanding the properties of trigonometric functions, identifying reference angles, and knowing the signs of the functions in different quadrants. By following these steps, you can confidently find the solutions to a wide range of trigonometric problems.

Expressing the Solutions

To summarize, we've found the solutions to the trigonometric equation cos(θ) = -1/2 over the interval [0°, 360°). We determined that the reference angle is 60°, and since cosine is negative in the second and third quadrants, we found the angles in those quadrants that have a cosine value of -1/2. These angles are 120° and 240°.

Therefore, the final solutions to the equation cos(θ) = -1/2 within the given interval are:

θ = 120° and θ = 240°

These are the only two angles within the interval [0°, 360°) that satisfy the equation. It’s always a good practice to verify your solutions by plugging them back into the original equation. If you calculate cos(120°) and cos(240°), you will find that both values are indeed -1/2.

Expressing the solutions clearly and accurately is an important part of the problem-solving process in mathematics. It demonstrates your understanding of the problem and your ability to communicate the results effectively. In this case, we have identified the specific angles that satisfy the equation, and we have presented them in a clear and concise manner.

Understanding how to solve trigonometric equations like this one is a fundamental skill in trigonometry. By mastering the steps involved – finding the reference angle, determining the quadrants, and identifying the solutions – you'll be well-equipped to tackle more complex trigonometric problems. Remember to always check your solutions and ensure they make sense within the context of the problem.

Conclusion

In this guide, we have walked through the process of solving the trigonometric equation cos(θ) = -1/2 over the interval [0°, 360°). We began by understanding the cosine function and its relationship to the unit circle. We then learned how to find the reference angle, which is a crucial step in solving trigonometric equations. Next, we identified the quadrants where the cosine function is negative, allowing us to narrow down the possible solutions. Finally, we determined the specific angles within the given interval that satisfy the equation, which are 120° and 240°.

This step-by-step approach can be applied to a variety of trigonometric equations, making it a valuable skill for anyone studying mathematics, physics, or engineering. By understanding the underlying principles and practicing these techniques, you can confidently solve trigonometric problems and deepen your understanding of trigonometry.

Remember, the key to success in trigonometry is a strong foundation in the basics. This includes understanding the unit circle, the trigonometric functions, and the relationships between angles and their trigonometric values. With practice and patience, you can master these concepts and excel in your studies.

For further exploration and practice with trigonometric functions, you might find the resources available at Khan Academy's Trigonometry section helpful.