Solving 1 - Cos Θ = 1/2 Trigonometric Equation On [0, 2π)
In the realm of trigonometry, solving equations is a fundamental skill. This article delves into the process of solving the trigonometric equation 1 - cos θ = 1/2 within the interval [0, 2π). We'll break down the steps, explain the underlying concepts, and provide a clear understanding of how to arrive at the solutions. Whether you're a student grappling with trigonometry or simply seeking to enhance your mathematical prowess, this guide will equip you with the knowledge and confidence to tackle similar problems.
Understanding Trigonometric Equations
Trigonometric equations are equations that involve trigonometric functions such as sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Solving these equations entails finding the values of the angle θ that satisfy the equation within a specified interval. The interval [0, 2π) represents one complete revolution around the unit circle, where 0 corresponds to 0 degrees and 2π corresponds to 360 degrees. The solutions to trigonometric equations often involve multiple angles due to the periodic nature of trigonometric functions. Understanding the unit circle and the properties of trigonometric functions is crucial for solving these equations effectively.
Isolating the Cosine Function
To begin solving the equation 1 - cos θ = 1/2, our initial step involves isolating the cosine function. This means we need to manipulate the equation to get cos θ by itself on one side. We start by subtracting 1 from both sides of the equation:
1 - cos θ - 1 = 1/2 - 1
This simplifies to:
-cos θ = -1/2
Next, we multiply both sides by -1 to eliminate the negative sign in front of the cosine term:
(-1) * (-cos θ) = (-1) * (-1/2)
This gives us:
cos θ = 1/2
Now that we have isolated the cosine function, we can proceed to find the angles θ that satisfy this condition. This is a crucial step because it allows us to focus solely on the values where the cosine function equals 1/2. The next phase involves identifying the reference angles and then determining the quadrants where cosine is positive.
Identifying Reference Angles
The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. It helps us find solutions in different quadrants. To find the reference angle for cos θ = 1/2, we recall the special right triangles, specifically the 30-60-90 triangle. In this triangle, the cosine of 60 degrees (or π/3 radians) is 1/2. Therefore, the reference angle is π/3. Knowing the reference angle is essential because it allows us to determine the angles in other quadrants that have the same cosine value. The cosine function is positive in the first and fourth quadrants, so we will look for angles in these quadrants.
Determining Quadrants Where Cosine is Positive
Cosine is positive in the first and fourth quadrants. In the first quadrant, the angle is the same as the reference angle. So, one solution is θ = π/3. In the fourth quadrant, the angle is 2π minus the reference angle. Therefore, the other solution is θ = 2π - π/3. To find this angle, we need to find a common denominator:
θ = (2π * 3)/3 - π/3
θ = 6π/3 - π/3
θ = 5π/3
So, the two angles in the interval [0, 2π) where cos θ = 1/2 are π/3 and 5π/3. These solutions correspond to the angles where the x-coordinate on the unit circle is 1/2. Identifying the quadrants is a critical step because it ensures we find all possible solutions within the given interval.
Finding Solutions in the Interval [0, 2π)
Now that we know the reference angle and the quadrants where cosine is positive, we can find the solutions within the interval [0, 2π). As determined earlier, the angles are:
- θ = π/3 (First Quadrant)
- θ = 5π/3 (Fourth Quadrant)
These are the only two angles within the interval [0, 2π) that satisfy the equation cos θ = 1/2. To verify these solutions, we can substitute them back into the original equation:
For θ = π/3:
1 - cos(π/3) = 1 - 1/2 = 1/2
For θ = 5π/3:
1 - cos(5π/3) = 1 - 1/2 = 1/2
Both solutions satisfy the original equation, confirming our results. Understanding how to find all solutions within the specified interval is crucial for mastering trigonometric equations. This involves not only identifying the reference angle but also applying the properties of trigonometric functions in different quadrants.
Conclusion
In summary, the solutions to the equation 1 - cos θ = 1/2 on the interval [0, 2π) are θ = π/3 and θ = 5π/3. We arrived at these solutions by first isolating the cosine function, then identifying the reference angle, and finally determining the angles in the appropriate quadrants. Solving trigonometric equations requires a solid understanding of trigonometric functions, the unit circle, and the properties of angles in different quadrants. By following a systematic approach, you can confidently solve a wide range of trigonometric equations. This detailed walkthrough should provide a strong foundation for further exploration of trigonometric concepts and problem-solving.
Choosing the Correct Answer:
Based on our calculations, the correct answer is:
D. π/3, 5π/3
This article provides a comprehensive guide to solving the trigonometric equation 1 - cos θ = 1/2 on the interval [0, 2π). By understanding the steps and concepts outlined, you can enhance your ability to solve similar trigonometric problems.
