Solving 2cos(πθ/4) = 1 On [0, 2π) A Step-by-Step Guide

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In this comprehensive guide, we will delve into the process of finding the exact solutions for the trigonometric equation 2cos(πθ/4) = 1 within the specified interval of [0, 2π). This problem falls under the domain of trigonometry, a branch of mathematics that explores the relationships between angles and sides of triangles. Specifically, we will be working with the cosine function, one of the fundamental trigonometric functions.

Understanding Trigonometric Equations

Before we dive into the solution, let's establish a clear understanding of trigonometric equations. These equations involve trigonometric functions (such as sine, cosine, tangent) and aim to find the values of the variable (in our case, θ) that satisfy the equation. Solving trigonometric equations often involves using inverse trigonometric functions, understanding the periodicity of trigonometric functions, and considering the specified interval for solutions.

Trigonometric equations are fundamental in various fields, including physics, engineering, and computer science. They help model periodic phenomena such as oscillations, waves, and rotations. Mastering the techniques for solving these equations is crucial for anyone pursuing a career in these areas.

Our specific equation, 2cos(πθ/4) = 1, involves the cosine function with a modified argument (πθ/4). The argument modification affects the period of the cosine function, which we will need to consider when finding the solutions. The interval [0, 2π) restricts the solutions we seek to angles within one full revolution of the unit circle.

Step-by-Step Solution

Let's break down the solution process into manageable steps:

1. Isolate the Cosine Function

The first step is to isolate the cosine function on one side of the equation. We can achieve this by dividing both sides of the equation by 2:

cos(πθ/4) = 1/2

This simplified equation allows us to focus solely on the cosine function and its argument.

Isolating the trigonometric function is a crucial step in solving any trigonometric equation. It allows us to directly apply the inverse trigonometric function and find the reference angles. Without isolating the function, it becomes significantly more challenging to determine the solutions.

2. Find the Reference Angle

The reference angle is the acute angle formed between the terminal side of the angle (πθ/4 in this case) and the x-axis. To find the reference angle, we need to determine the angle whose cosine is 1/2. Recall the unit circle and the special angles. We know that:

cos(π/3) = 1/2

Therefore, the reference angle is π/3.

The reference angle is a fundamental concept in trigonometry. It allows us to relate angles in different quadrants to a standard acute angle. By knowing the reference angle and the sign of the trigonometric function in a particular quadrant, we can determine all possible solutions within a given interval.

3. Identify Quadrants

Since the cosine function is positive (1/2) in the first and fourth quadrants, we need to find angles in these quadrants that have a reference angle of π/3.

Understanding the signs of trigonometric functions in different quadrants is crucial for finding all possible solutions. The acronym ASTC (All, Sine, Tangent, Cosine) is a helpful mnemonic to remember which trigonometric functions are positive in each quadrant.

  • Quadrant I: All trigonometric functions are positive.
  • Quadrant II: Sine is positive.
  • Quadrant III: Tangent is positive.
  • Quadrant IV: Cosine is positive.

4. Determine the Angles

  • Quadrant I: The angle in the first quadrant with a reference angle of π/3 is simply π/3 itself.

    So, one possible solution is:

    πθ/4 = π/3

  • Quadrant IV: The angle in the fourth quadrant with a reference angle of π/3 is 2π - π/3 = 5π/3.

    So, another possible solution is:

    πθ/4 = 5π/3

Finding the angles in the correct quadrants is a key step in solving trigonometric equations. It ensures that we capture all possible solutions within the given interval. Visualizing the unit circle can be extremely helpful in determining these angles.

5. Solve for θ

Now we need to solve the equations we obtained in the previous step for θ:

  • πθ/4 = π/3

    Multiply both sides by 4/π:

    θ = (π/3) * (4/π) = 4/3

  • πθ/4 = 5π/3

    Multiply both sides by 4/π:

    θ = (5π/3) * (4/π) = 20/3

Solving for the variable is the final algebraic step in finding the solutions. It involves isolating the variable using appropriate algebraic operations.

6. Consider the Periodicity

The cosine function is periodic with a period of 2π. However, our function is cos(πθ/4), so we need to consider the modified period. The period of cos(Bθ) is 2π/|B|. In our case, B = π/4, so the period is 2π / (π/4) = 8.

This means that the solutions repeat every 8 units. We need to check if adding or subtracting multiples of 8 from our current solutions yields additional solutions within the interval [0, 2π).

The periodicity of trigonometric functions is a crucial aspect to consider when finding all possible solutions. Because trigonometric functions repeat their values over regular intervals, we need to account for this periodicity to ensure that we do not miss any solutions.

7. Check for Solutions in the Interval [0, 2π)

Our solutions so far are θ = 4/3 and θ = 20/3. We need to verify if these solutions, and any solutions generated by adding or subtracting multiples of the period, lie within the interval [0, 2π).

  • θ = 4/3

    This solution is within the interval [0, 2π) since 4/3 is approximately 1.33, which is less than 2π (approximately 6.28).

  • θ = 20/3

    This solution is outside the interval [0, 2π) since 20/3 is approximately 6.67, which is greater than 2π.

Now, let's consider the periodicity. We need to determine if there are any other solutions within the interval by adding or subtracting multiples of 8. Since our interval is [0, 2π), which is approximately [0, 6.28], adding or subtracting 8 from 4/3 or 20/3 will result in values outside this interval. Therefore, there are no other solutions within the specified range.

Checking the interval is a critical step in the solution process. It ensures that we only include solutions that are relevant to the given problem. Failing to check the interval can lead to extraneous solutions, which are values that satisfy the equation but not the given conditions.

Final Solution

Therefore, the exact solution for the equation 2cos(πθ/4) = 1 on the interval [0, 2π) is:

θ = 4/3

Alternative Solutions and Considerations

While we have found the primary solution within the interval [0, 2π), it's beneficial to consider alternative approaches and potential complexities that might arise in similar problems.

General Solutions

The general solutions to the equation can be expressed by considering the periodicity of the cosine function. The general solutions are given by:

πθ/4 = 2nπ ± π/3, where n is an integer.

Solving for θ, we get:

θ = 8n ± 4/3, where n is an integer.

These general solutions represent all possible solutions to the equation, not just those within the interval [0, 2π). To find specific solutions within a given interval, we can substitute different integer values for n and check if the resulting θ values fall within the interval.

Graphical Approach

Another way to visualize and solve trigonometric equations is by using a graphical approach. We can graph the function y = 2cos(πθ/4) and the horizontal line y = 1. The points of intersection of these graphs represent the solutions to the equation 2cos(πθ/4) = 1. By observing the graph within the interval [0, 2π), we can identify the solutions.

A graphical approach can be particularly helpful for complex trigonometric equations where algebraic solutions are difficult to obtain. It provides a visual representation of the solutions and can help in understanding the behavior of the trigonometric functions.

Complex Arguments

In some cases, the argument of the trigonometric function might involve more complex expressions. For instance, we might encounter equations like cos(πθ²/4) = 1/2. Solving such equations often requires careful algebraic manipulation and consideration of the domain of the function.

Conclusion

Finding exact solutions to trigonometric equations requires a combination of algebraic manipulation, understanding of trigonometric functions, and consideration of intervals and periodicity. By following a systematic approach, we can solve a wide range of trigonometric equations. In this article, we meticulously solved the equation 2cos(πθ/4) = 1 within the interval [0, 2π), demonstrating the key steps and concepts involved. Remember to always isolate the trigonometric function, find the reference angle, identify the relevant quadrants, and consider the periodicity to arrive at the correct solutions.

This comprehensive guide has provided a detailed explanation of how to solve trigonometric equations of this type. By understanding the concepts and techniques discussed, you will be well-equipped to tackle similar problems and deepen your understanding of trigonometry.