Solving 2cos(θ) - √2 = 0 Trigonometric Equation In [0, 2π)

Jul 8, 2025 by Admin 59 views

Solving Trigonometric Equations: Finding Solutions for 2cos(θ) - √2 = 0 in the Interval [0, 2π)

In the realm of trigonometric equations, finding solutions within a specified interval is a fundamental task. This article delves into the process of solving the equation 2cos(θ) - √2 = 0 within the interval [0, 2π). We'll explore the underlying principles, step-by-step solution, and the significance of the obtained results. Trigonometric equations, at their core, involve trigonometric functions such as sine, cosine, and tangent, and our goal is to determine the angles that satisfy the given equation. The interval [0, 2π) represents a full circle in radians, which is a standard range for finding solutions to trigonometric equations. Understanding how to solve these equations is crucial in various fields, including physics, engineering, and computer graphics, where periodic phenomena are modeled using trigonometric functions. This process involves algebraic manipulation, understanding the unit circle, and applying the inverse trigonometric functions to isolate the variable. In the specific equation we are addressing, the cosine function is the central element, and we aim to find the angles at which the cosine function yields a value that satisfies the equation. This involves understanding the cosine function's behavior in the given interval and identifying the specific angles that correspond to the required value. By carefully considering the properties of the cosine function and the interval in question, we can systematically find all solutions to the equation and gain a deeper understanding of the underlying trigonometric principles. This exercise not only enhances our mathematical skills but also provides a foundation for more complex problem-solving in related fields.

Understanding the Basics of Trigonometric Equations

Before diving into the solution, it's essential to grasp the basics of trigonometric equations. These equations involve trigonometric functions like sine, cosine, tangent, and their reciprocals. Solving them means finding the angles (usually in radians or degrees) that make the equation true. The unit circle is a powerful tool for visualizing trigonometric values and their relationships. It's a circle with a radius of 1 centered at the origin of a coordinate plane. Angles are measured counterclockwise from the positive x-axis, and the coordinates of a point on the circle correspond to the cosine and sine of that angle. The cosine function, specifically, represents the x-coordinate of a point on the unit circle, while the sine function represents the y-coordinate. Understanding the unit circle allows us to quickly determine the values of trigonometric functions for common angles such as 0, π/6, π/4, π/3, π/2, and their multiples. This visual representation helps us to identify multiple solutions within the interval [0, 2π), as trigonometric functions are periodic. The periodicity of these functions means that they repeat their values at regular intervals, leading to multiple solutions for a given equation. For instance, the cosine function has a period of 2π, which means that cos(θ) = cos(θ + 2π) for any angle θ. Therefore, when solving trigonometric equations, it's crucial to consider all possible solutions within the specified interval. By combining algebraic techniques with a solid understanding of the unit circle and the periodic nature of trigonometric functions, we can effectively solve a wide range of trigonometric equations and apply these skills to various practical problems.

Step-by-Step Solution for 2cos(θ) - √2 = 0

Let's solve the equation 2cos(θ) - √2 = 0 step by step. Our goal is to isolate cos(θ) and then find the angles θ in the interval [0, 2π) that satisfy the equation.

Isolate cos(θ):
- Add √2 to both sides of the equation:
```
2cos(θ) = √2
```
- Divide both sides by 2:
```
cos(θ) = √2 / 2
```
Identify angles with cos(θ) = √2 / 2:
- Recall the unit circle and the angles where the x-coordinate (cosine value) is √2 / 2. These angles are in the first and fourth quadrants.
- The reference angle is π/4 (45 degrees), as cos(π/4) = √2 / 2.
Find solutions in the interval [0, 2π):
- In the first quadrant, θ = π/4 is a solution.
- In the fourth quadrant, the angle is 2π - π/4 = 7π/4.

Therefore, the solutions to the equation 2cos(θ) - √2 = 0 in the interval [0, 2π) are π/4 and 7π/4. This process involves a combination of algebraic manipulation and understanding the unit circle. By isolating the trigonometric function and then using our knowledge of the unit circle, we can identify the angles that satisfy the equation within the given interval. The reference angle plays a crucial role in finding solutions in different quadrants, as it allows us to relate the given trigonometric value to a known angle. In this case, the reference angle of π/4 helped us to find solutions in both the first and fourth quadrants. This systematic approach can be applied to solve a wide variety of trigonometric equations, making it a fundamental skill in trigonometry and related fields.

Verifying the Solutions

To ensure the correctness of our solutions, we need to verify that π/4 and 7π/4 indeed satisfy the original equation 2cos(θ) - √2 = 0. This verification process is crucial in mathematical problem-solving, as it helps to identify any potential errors or inconsistencies in our calculations. By substituting each solution back into the original equation, we can confirm whether the equation holds true. This step not only provides confidence in our answer but also reinforces our understanding of the trigonometric functions and their properties. The process involves evaluating the cosine function at each solution and then performing the arithmetic operations to check if the equation is satisfied. This can be done manually or with the aid of a calculator, ensuring accuracy in the verification process. Furthermore, this verification step can also help to identify any extraneous solutions that may have arisen during the algebraic manipulation of the equation. Extraneous solutions are values that satisfy the transformed equation but not the original equation. By carefully verifying each solution, we can ensure that our final answer consists only of the true solutions to the original problem.

For θ = π/4:
- Substitute θ = π/4 into the equation:
```
2cos(π/4) - √2 = 2(√2 / 2) - √2 = √2 - √2 = 0
```
- The equation holds true.
For θ = 7π/4:
- Substitute θ = 7π/4 into the equation:
```
2cos(7π/4) - √2 = 2(√2 / 2) - √2 = √2 - √2 = 0
```
- The equation holds true.

Both solutions, π/4 and 7π/4, satisfy the original equation. This confirms that our solutions are correct and valid within the given interval. The verification process not only solidifies our understanding of the problem-solving steps but also provides a sense of confidence in the accuracy of the final answer. It is a crucial step in any mathematical problem, ensuring that the solutions obtained are indeed the correct ones.

Visualizing Solutions on the Unit Circle

The unit circle is an invaluable tool for visualizing trigonometric solutions. It provides a geometric representation of trigonometric functions and their values at different angles. By plotting the solutions on the unit circle, we can gain a deeper understanding of their relationship and their positions within the interval [0, 2π). This visual representation helps to reinforce the concept that trigonometric functions are periodic and have multiple solutions within a given interval. The unit circle, with its radius of 1, allows us to easily identify the cosine and sine values for any angle, as they correspond to the x and y coordinates of the point on the circle. In the case of the equation 2cos(θ) - √2 = 0, we found the solutions to be π/4 and 7π/4. These angles can be easily located on the unit circle, where the x-coordinate (cosine value) is √2 / 2. By visualizing these solutions, we can see that they are symmetrically positioned with respect to the x-axis, reflecting the property that cosine is an even function. Furthermore, the unit circle helps to illustrate the periodic nature of the cosine function, as we can see that adding multiples of 2π to these solutions would result in the same point on the circle, representing the same cosine value. This visual aid not only enhances our understanding of the solutions but also provides a foundation for solving more complex trigonometric problems.

Draw a unit circle (a circle with a radius of 1 centered at the origin).
Mark the angles π/4 and 7π/4 on the circle.
Observe that these angles correspond to the points where the x-coordinate is √2 / 2, which is the value of cos(θ) we found earlier.

Visualizing the solutions on the unit circle reinforces the concept that there can be multiple solutions to a trigonometric equation within the interval [0, 2π). It also helps to understand the symmetry and periodicity of trigonometric functions. This graphical representation adds another layer of understanding to the algebraic solution, making the concepts more intuitive and memorable.

Common Mistakes and How to Avoid Them

When solving trigonometric equations, several common mistakes can occur. Being aware of these pitfalls and understanding how to avoid them is crucial for achieving accurate solutions. One common mistake is forgetting to consider all possible solutions within the given interval. Trigonometric functions are periodic, meaning they repeat their values at regular intervals. Therefore, it's essential to check for solutions in all quadrants and to account for the periodicity of the functions. Another mistake is incorrectly applying inverse trigonometric functions. The inverse trigonometric functions, such as arcsin, arccos, and arctan, have restricted ranges, which means they only return one solution. To find all solutions, it's necessary to use the reference angle and consider the quadrants where the trigonometric function has the same sign. Additionally, algebraic errors can occur during the manipulation of the equation, such as incorrect factoring or simplification. It's crucial to double-check each step to ensure accuracy. Furthermore, extraneous solutions can sometimes arise during the solving process, particularly when squaring both sides of an equation or multiplying by a trigonometric function. These extraneous solutions must be identified and discarded by verifying the solutions in the original equation. By being mindful of these common mistakes and implementing strategies to avoid them, such as careful algebraic manipulation, thorough consideration of the unit circle, and verification of solutions, we can significantly improve our accuracy and confidence in solving trigonometric equations.

Forgetting the periodicity of trigonometric functions: Always consider all possible solutions within the given interval. Trigonometric functions repeat their values, so there might be more than one solution.
Incorrectly using inverse trigonometric functions: Inverse trigonometric functions have restricted ranges. Use the unit circle to find all solutions.
Algebraic errors: Double-check your algebraic manipulations to avoid mistakes.
Extraneous solutions: Verify your solutions in the original equation to eliminate any extraneous solutions.

By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in solving trigonometric equations.

Conclusion

In conclusion, solving the trigonometric equation 2cos(θ) - √2 = 0 in the interval [0, 2π) involves a systematic approach that combines algebraic manipulation, understanding the unit circle, and verifying the solutions. We successfully found two solutions, π/4 and 7π/4, which satisfy the equation within the given interval. This process highlights the importance of understanding the periodic nature of trigonometric functions and the use of the unit circle as a visual aid. By following a step-by-step method, we can isolate the trigonometric function, identify the reference angle, and find all solutions within the specified interval. Verification of the solutions is crucial to ensure accuracy and to eliminate any extraneous solutions. Furthermore, visualizing the solutions on the unit circle provides a deeper understanding of their relationship and their positions within the trigonometric context. By avoiding common mistakes, such as forgetting the periodicity of trigonometric functions or incorrectly using inverse trigonometric functions, we can improve our problem-solving skills and achieve accurate results. This exercise not only enhances our mathematical abilities but also provides a foundation for tackling more complex problems in various fields where trigonometric functions are applied. The ability to solve trigonometric equations is a valuable skill in mathematics and its applications, and mastering this skill requires a combination of conceptual understanding and practical application.

The solutions to the equation 2cos(θ) - √2 = 0 in the interval [0, 2π) are: π/4, 7π/4.