Solving Tan Θ - 1 = 0 Find All Solutions In [0, 2π)

In the realm of trigonometry, solving equations is a fundamental skill. Trigonometric equations often arise in various contexts, from physics and engineering to computer graphics and music theory. Mastering the techniques to solve these equations is crucial for a solid understanding of mathematical principles and their applications in the real world. In this comprehensive guide, we will delve into the process of solving the specific trigonometric equation tan θ - 1 = 0 within the interval [0, 2π). This interval represents a full revolution around the unit circle, and our goal is to identify all angles θ within this range that satisfy the given equation. We will explore the properties of the tangent function, its behavior within the unit circle, and the strategies for finding solutions. By the end of this guide, you will have a clear understanding of how to solve this particular equation and a broader appreciation for the techniques involved in solving trigonometric equations in general.

To effectively tackle the equation tan θ - 1 = 0, it is essential to have a firm grasp of the tangent function. The tangent function, denoted as tan θ, is one of the fundamental trigonometric functions, alongside sine (sin θ) and cosine (cos θ). It is defined as the ratio of the sine of an angle to its cosine: tan θ = sin θ / cos θ. Geometrically, on the unit circle, tan θ represents the slope of the line that passes through the origin and the point on the circle corresponding to the angle θ. The tangent function has several key properties that are crucial for solving trigonometric equations. First, it is a periodic function with a period of π, meaning that its values repeat every π radians. This implies that if θ is a solution to tan θ - 1 = 0, then θ + nπ, where n is an integer, will also be a solution. However, we are specifically interested in solutions within the interval [0, 2π), so we need to consider only the solutions that fall within this range. Second, the tangent function has vertical asymptotes at angles where the cosine function is zero, i.e., at θ = π/2 + nπ, where n is an integer. This is because division by zero is undefined. These asymptotes indicate that the tangent function approaches infinity or negative infinity as θ approaches these values. Third, the tangent function is positive in the first and third quadrants of the unit circle and negative in the second and fourth quadrants. This information is valuable in determining the possible quadrants where the solutions to our equation might lie. Understanding these properties of the tangent function will greatly aid us in finding all solutions to the equation tan θ - 1 = 0 within the interval [0, 2π).

Now, let's focus on solving the equation tan θ - 1 = 0. The first step is to isolate the tangent function. By adding 1 to both sides of the equation, we get tan θ = 1. This equation asks: for what angles θ is the tangent function equal to 1? To answer this, we need to recall the definition of the tangent function and its behavior on the unit circle. We know that tan θ = sin θ / cos θ. Therefore, tan θ = 1 when sin θ = cos θ. This occurs at angles where the x and y coordinates of the point on the unit circle are equal. From our knowledge of the unit circle, we know that this happens at θ = π/4 (45 degrees) in the first quadrant. At this angle, both the sine and cosine are equal to √2/2, and their ratio is 1. However, this is not the only solution within the interval [0, 2π). Since the tangent function has a period of π, we can find another solution by adding π to the first solution. This gives us θ = π/4 + π = 5π/4. This angle lies in the third quadrant, where both sine and cosine are negative, but their ratio is still 1. Adding another π to 5π/4 would give us 9π/4, which is greater than 2π and therefore outside our interval of interest. Thus, the solutions to the equation tan θ - 1 = 0 in the interval [0, 2π) are θ = π/4 and θ = 5π/4. These are the only two angles within a full revolution of the unit circle where the tangent function equals 1. We have successfully identified all solutions by utilizing our understanding of the tangent function, its properties, and its behavior on the unit circle.

To further solidify our understanding, let's discuss the concept of general solutions and how the unit circle plays a crucial role in finding them. We found that the solutions to tan θ - 1 = 0 within the interval [0, 2π) are θ = π/4 and θ = 5π/4. However, due to the periodic nature of the tangent function, there are infinitely many solutions to this equation. The general solution represents all possible solutions, not just those within a specific interval. Since the tangent function has a period of π, we can express the general solution as θ = π/4 + nπ, where n is an integer. This formula generates all angles that have a tangent of 1. For example, if n = 0, we get θ = π/4; if n = 1, we get θ = 5π/4; if n = 2, we get θ = 9π/4, and so on. Similarly, if n is negative, we get solutions such as θ = -3π/4, which is coterminal with 5π/4. The unit circle is an invaluable tool for visualizing and understanding trigonometric functions and their solutions. It is 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 the point where the terminal side of the angle intersects the unit circle are (cos θ, sin θ). The tangent function, tan θ = sin θ / cos θ, can be visualized as the slope of the line connecting the origin to the point on the unit circle. When solving trigonometric equations, the unit circle helps us identify the angles that satisfy the given conditions. For tan θ = 1, we look for points on the unit circle where the slope of the line connecting the origin to the point is 1. This occurs at π/4 and 5π/4, as we found earlier. The unit circle provides a visual representation of the periodic nature of trigonometric functions and helps us understand how the general solution formula generates all possible solutions. By understanding the general solutions and leveraging the unit circle, we can confidently solve a wide range of trigonometric equations.

In the context of trigonometric equations, it is essential to express solutions in radians, especially when dealing with mathematical and scientific applications. Radians are a unit of angular measure defined such that an angle of one radian subtended at the center of a circle by an arc equal in length to the radius of the circle. The radian measure is a dimensionless quantity, making it the preferred unit for mathematical analysis and calculations involving trigonometric functions. Unlike degrees, which are based on an arbitrary division of the circle into 360 parts, radians are directly related to the geometry of the circle. The relationship between radians and degrees is given by π radians = 180 degrees. This conversion factor allows us to switch between the two units. For example, 90 degrees is equivalent to π/2 radians, 180 degrees is equivalent to π radians, and 360 degrees is equivalent to 2π radians. In our problem, we were asked to find the solutions to tan θ - 1 = 0 in the interval [0, 2π) and express them in radians in terms of π. We found the solutions to be θ = π/4 and θ = 5π/4. These solutions are already expressed in radians and in terms of π, as requested. Expressing solutions in radians is crucial because it aligns with the mathematical definitions of trigonometric functions and their relationships to other mathematical concepts, such as calculus and complex numbers. When working with trigonometric equations and functions, it is best practice to use radians unless otherwise specified. By consistently using radians, we ensure that our calculations and interpretations are mathematically sound and consistent with established conventions.

The core of solving trigonometric equations lies in identifying all solutions within the specified interval. In our case, the interval is [0, 2π), which represents a full revolution around the unit circle. To ensure we capture all solutions, we need to systematically analyze the behavior of the trigonometric function in question and consider its periodic nature. For the equation tan θ - 1 = 0, we have already determined that tan θ = 1. We know that the tangent function is positive in the first and third quadrants of the unit circle. Therefore, we expect to find solutions in these quadrants. The reference angle, which is the acute angle formed by the terminal side of the angle and the x-axis, for tan θ = 1 is π/4. This means that one solution is θ = π/4, which lies in the first quadrant. To find the solution in the third quadrant, we add π to the reference angle: θ = π/4 + π = 5π/4. This gives us another solution within the interval [0, 2π). It is essential to check if there are any other solutions within the interval. Since the tangent function has a period of π, adding another π to 5π/4 would give us 9π/4, which is greater than 2π and therefore outside our interval. Similarly, subtracting π from π/4 would give us -3π/4, which is less than 0 and also outside our interval. This confirms that π/4 and 5π/4 are the only solutions within the interval [0, 2π). When identifying all solutions, it is crucial to consider the periodicity of the trigonometric function and the boundaries of the interval. By systematically analyzing the function's behavior and using the unit circle as a visual aid, we can confidently find all solutions and avoid missing any. This thorough approach is key to mastering the art of solving trigonometric equations.

Once we have identified all the solutions to a trigonometric equation within a given interval, it is important to express the solution set clearly and concisely. The solution set is a collection of all the angles that satisfy the equation within the specified interval. In our case, the equation is tan θ - 1 = 0, and the interval is [0, 2π). We found that the solutions are θ = π/4 and θ = 5π/4. There are several ways to express the solution set. One common method is to use set notation. In set notation, the solution set is written as {π/4, 5π/4}. This notation explicitly lists all the elements in the set, separated by commas and enclosed in curly braces. Another way to express the solution set is to write it as a list, separating the solutions with commas. This method is often used when the problem statement asks for the solutions to be separated by commas, as in our case. So, we can write the solution as θ = π/4, 5π/4. It is crucial to follow the instructions in the problem statement regarding how to express the solution set. Some problems may require the solutions to be expressed in a specific format, such as set notation or a comma-separated list. Additionally, the solutions should be expressed in the correct units, which in our case are radians. When expressing the solution set, it is also important to ensure that all solutions within the interval are included and that no extraneous solutions are present. This requires a careful and systematic approach to solving the equation and verifying the solutions. By expressing the solution set clearly and accurately, we effectively communicate the results of our work and demonstrate our understanding of the problem.

In conclusion, we have successfully solved the trigonometric equation tan θ - 1 = 0 within the interval [0, 2π). By understanding the properties of the tangent function, utilizing the unit circle, and systematically identifying all solutions, we found that θ = π/4 and θ = 5π/4 are the only angles within the given interval that satisfy the equation. We also discussed the concept of general solutions and how they represent all possible solutions due to the periodic nature of the tangent function. Furthermore, we emphasized the importance of expressing solutions in radians and expressing the solution set clearly and concisely. Solving trigonometric equations is a fundamental skill in mathematics, with applications in various fields. The techniques and concepts discussed in this guide provide a solid foundation for tackling more complex trigonometric problems. By mastering these skills, you will be well-equipped to solve a wide range of trigonometric equations and apply them to real-world scenarios. Remember to always consider the properties of the trigonometric functions, utilize the unit circle, and systematically identify all solutions within the specified interval. With practice and a solid understanding of the underlying principles, you can confidently solve trigonometric equations and expand your mathematical expertise.