Solving The Inequality √sin(x) (sin(x) + 1/2) > 0 A Comprehensive Guide
This article delves into the mathematical inequality √sin(x) (sin(x) + 1/2) > 0, providing a comprehensive analysis and discussion of its solution. We will explore the underlying concepts, break down the problem step-by-step, and arrive at a clear understanding of the conditions under which this inequality holds true. This exploration involves understanding the domain of the square root function, the range of the sine function, and how these interact to determine the overall solution. We aim to provide a detailed explanation suitable for students and enthusiasts alike, fostering a deeper appreciation for the intricacies of mathematical inequalities.
Understanding the Components of the Inequality
Before we dive into solving the inequality, it's crucial to understand the individual components that make it up. Our inequality is √sin(x) (sin(x) + 1/2) > 0, and it features two primary elements: the square root of the sine function, √sin(x), and the expression (sin(x) + 1/2). Each of these components has its own unique characteristics and constraints that influence the overall behavior of the inequality. Let's examine each part in detail.
The Square Root of Sine: √sin(x)
The first component we need to understand is √sin(x), the square root of the sine function. The square root function, by its very nature, imposes a significant restriction: it is only defined for non-negative values. This means that the expression inside the square root, in this case, sin(x), must be greater than or equal to zero. Mathematically, this can be expressed as sin(x) ≥ 0. This is a critical condition because it limits the possible values of x for which the entire expression is even defined. The sine function is non-negative in the first and second quadrants of the unit circle, which translates to the intervals [0, π] and their periodic repetitions. Therefore, x must lie within these intervals for √sin(x) to be a real number.
Furthermore, the square root function itself is always non-negative. This means that √sin(x) will always be greater than or equal to zero. It will be equal to zero when sin(x) = 0, which occurs at integer multiples of π (i.e., x = nπ, where n is an integer). When sin(x) is positive, √sin(x) will also be positive. This behavior is crucial for understanding when the overall inequality will be satisfied. We need to ensure not only that √sin(x) is defined but also that its value contributes to making the entire expression greater than zero.
The implications of the square root are profound. It restricts the domain of our problem significantly, forcing us to consider only those x values where sin(x) is not negative. This initial restriction is a cornerstone in solving the inequality, as it narrows down the possibilities and guides our subsequent steps. Ignoring this constraint could lead to incorrect solutions, emphasizing the importance of thoroughly understanding the function's properties.
Analyzing the Expression: sin(x) + 1/2
The second crucial component of our inequality is the expression sin(x) + 1/2. This expression involves the sine function, which oscillates between -1 and 1, and a constant value of 1/2. Understanding how these two elements interact is key to determining when this expression is positive, negative, or zero. To do this effectively, we need to consider the range of the sine function and how adding 1/2 shifts that range.
The sine function, sin(x), has a well-defined range: it varies between -1 and 1, inclusive. This means that for any value of x, the output of sin(x) will always fall within the interval [-1, 1]. Now, let's consider what happens when we add 1/2 to this function. Adding a constant value to a function shifts its entire graph vertically. In this case, adding 1/2 to sin(x) shifts the entire sine wave upward by 1/2 unit. Consequently, the range of the expression sin(x) + 1/2 becomes [-1/2, 3/2]. This shifted range is crucial because it tells us the possible values that the expression can take.
Our primary concern is to determine when sin(x) + 1/2 is greater than zero. Since the range of the expression is [-1/2, 3/2], we know that the expression is greater than zero for a significant portion of the sine function's cycle. To find the exact intervals where sin(x) + 1/2 > 0, we need to solve the inequality sin(x) > -1/2. This involves finding the angles x for which the sine function is greater than -1/2. We can visualize this on the unit circle by identifying the regions where the y-coordinate (which represents the sine value) is greater than -1/2. These regions correspond to specific intervals of angles, which will ultimately contribute to the solution of our overall inequality.
The solutions to sin(x) > -1/2 can be found using trigonometric principles and the unit circle. The reference angles and the periodic nature of the sine function play a vital role in determining the complete set of solutions. Specifically, we need to find the angles where the sine function dips below -1/2 and exclude those intervals from our solution set. This analysis is essential because it identifies another set of constraints on x that must be considered in conjunction with the constraints imposed by the square root function. Understanding the behavior of sin(x) + 1/2 is pivotal in piecing together the complete picture of our inequality's solution.
Solving the Inequality √sin(x) (sin(x) + 1/2) > 0: A Step-by-Step Approach
Now that we have a solid understanding of the individual components of the inequality, √sin(x) (sin(x) + 1/2) > 0, we can proceed with solving it. This process involves combining the constraints imposed by the square root and the trigonometric expression to determine the intervals where the inequality holds true. We will break down the solution into a step-by-step approach to ensure clarity and accuracy.
Step 1: Domain Restriction from the Square Root
The first and most crucial step is to address the domain restriction imposed by the square root function. As we discussed earlier, the expression inside the square root, sin(x), must be non-negative for the square root to be defined in the real number system. This gives us the primary condition: sin(x) ≥ 0. This condition restricts our possible solutions to the intervals where the sine function is either positive or zero.
The sine function is non-negative in the first and second quadrants of the unit circle. This corresponds to the intervals [0, π] and their periodic repetitions. Mathematically, we can express this as:
2nπ ≤ x ≤ (2n + 1)π, where n is any integer.
This inequality defines the fundamental domain of our problem. We can only consider values of x that fall within these intervals. Any solution we find must adhere to this restriction. It's important to visualize this on the graph of the sine function; we are essentially focusing on the portions of the graph that lie above or on the x-axis. This initial restriction is a cornerstone in solving the inequality, as it significantly narrows down the possibilities and guides our subsequent steps. Ignoring this constraint could lead to incorrect solutions, emphasizing the importance of thoroughly understanding the function's properties.
Step 2: Analyzing the Expression sin(x) + 1/2
The next step is to analyze the second component of the inequality: sin(x) + 1/2. We need to determine when this expression is positive, as it will play a crucial role in satisfying the overall inequality. Recall that we are trying to solve √sin(x) (sin(x) + 1/2) > 0. Since √sin(x) is always non-negative within its domain, the sign of the entire expression depends on the sign of sin(x) + 1/2.
We need to solve the inequality sin(x) + 1/2 > 0, which is equivalent to sin(x) > -1/2. To find the intervals where this holds true, we can consider the unit circle. The sine function corresponds to the y-coordinate on the unit circle, so we are looking for angles where the y-coordinate is greater than -1/2. This occurs in the first, second, and part of the fourth quadrants. The angles where sin(x) = -1/2 are 7π/6 and 11π/6 in the interval [0, 2π]. Therefore, sin(x) > -1/2 when x is in the intervals (0, 7π/6) and (11π/6, 2π) within one period.
Considering the periodic nature of the sine function, we can generalize these intervals as follows:
2nπ - π/6 < x < 2nπ + 7π/6, where n is any integer.
This inequality provides us with the intervals where sin(x) + 1/2 is positive. However, we must remember that we also have the domain restriction from the square root function. We need to find the intersection of these two sets of intervals to determine the final solution.
Step 3: Combining the Conditions and Finding the Solution
Now we come to the critical step of combining the conditions we've derived from the square root and the trigonometric expression. We have two primary conditions:
- sin(x) ≥ 0, which implies 2nπ ≤ x ≤ (2n + 1)π
- sin(x) > -1/2, which implies 2nπ - π/6 < x < 2nπ + 7π/6
To satisfy the original inequality √sin(x) (sin(x) + 1/2) > 0, both √sin(x) and (sin(x) + 1/2) must be positive (since the product of two positive numbers is positive). √sin(x) is positive when sin(x) > 0 (and not just sin(x) ≥ 0, because if sin(x) = 0, the entire expression becomes zero, violating the strict inequality). Therefore, we need to find the intersection of the intervals where sin(x) > 0 and sin(x) > -1/2.
Let's analyze this within one period, say [0, 2π]. sin(x) > 0 in the interval (0, π), and sin(x) > -1/2 in the intervals (0, 7π/6) and (11π/6, 2π). The intersection of these intervals is (0, π). Note that we exclude x = 0 and x = π because at these points, sin(x) = 0, making √sin(x) = 0 and thus the entire expression equal to zero.
Extending this to all real numbers, we get the general solution:
2nπ < x < (2n + 1)π, where n is any integer.
This is the final solution to the inequality √sin(x) (sin(x) + 1/2) > 0. It represents the set of all x values for which the inequality holds true. This solution arises from the careful consideration of the domain restrictions imposed by the square root function and the analysis of the trigonometric expression. Each step in this process is crucial to ensure an accurate and complete solution.
Graphical Representation and Interpretation
To further enhance our understanding of the solution to the inequality √sin(x) (sin(x) + 1/2) > 0, it's incredibly helpful to visualize the functions involved graphically. A graphical representation allows us to see the behavior of the functions and the solution intervals in a clear and intuitive way. We can plot the function y = √sin(x) (sin(x) + 1/2) and identify the regions where the graph lies above the x-axis, as these correspond to the values of x for which the inequality is satisfied. This graphical approach not only confirms our analytical solution but also provides a deeper insight into the problem.
Plotting the Function y = √sin(x) (sin(x) + 1/2)
When plotting the function y = √sin(x) (sin(x) + 1/2), several key features become immediately apparent. First and foremost, we observe the domain restriction imposed by the square root function. The graph exists only where sin(x) ≥ 0, which corresponds to the intervals [0, π], [2π, 3π], and so on. This means the graph is defined in segments, with gaps in between where sin(x) is negative. These gaps visually represent the domain restriction we discussed earlier, reinforcing the importance of considering the square root's constraints.
Within the defined segments, we see the function oscillating, reflecting the periodic nature of the sine function. The peaks and troughs of the graph indicate the maximum and minimum values of the expression within each interval. More importantly, we can clearly see where the graph lies above the x-axis (y > 0) and where it touches or crosses the x-axis (y ≤ 0). The regions where the graph is above the x-axis visually represent the solutions to our inequality.
The graph starts at x = 0 with y = 0, rises to a positive value, and then returns to y = 0 at x = π. This pattern repeats for every interval of the form [2nπ, (2n + 1)π], where n is an integer. The fact that the graph touches the x-axis at multiples of π emphasizes why these points are excluded from our solution, as the inequality is strict (> rather than ≥). The shape of the curve within each interval provides a visual representation of how the product of √sin(x) and (sin(x) + 1/2) behaves, showing the interplay between these two components.
Interpreting the Graph in Relation to the Solution
The graphical representation directly confirms our analytical solution. The intervals where the graph of y = √sin(x) (sin(x) + 1/2) lies above the x-axis correspond precisely to the intervals we found algebraically: 2nπ < x < (2n + 1)π, where n is an integer. This visual confirmation is a powerful tool for understanding the solution. We can see why the solution is a series of open intervals, excluding the endpoints where the function equals zero.
The graph also helps to illustrate the role of the sin(x) + 1/2 term. While √sin(x) ensures that the function is only defined for non-negative values of sin(x), the sin(x) + 1/2 term influences the magnitude and sign of the expression. The graph shows how this term, combined with the square root, creates the oscillating pattern within the allowed intervals. It also highlights that even though sin(x) + 1/2 can be negative for some values of x, the square root ensures that we only consider the regions where the overall expression is either positive or zero.
In conclusion, the graphical representation is an invaluable tool for understanding and confirming the solution to the inequality √sin(x) (sin(x) + 1/2) > 0. It provides a visual context for the algebraic solution, making the concepts more accessible and intuitive. By plotting the function and interpreting its behavior, we gain a deeper appreciation for the interplay between the trigonometric and square root components of the inequality.
Common Mistakes and Pitfalls
When solving inequalities, especially those involving trigonometric functions and square roots, it's easy to make mistakes if careful attention isn't paid to the details. Understanding common pitfalls can help prevent errors and ensure a correct solution. Let's explore some of the most frequent mistakes encountered when dealing with inequalities like √sin(x) (sin(x) + 1/2) > 0.
Neglecting the Domain Restriction of the Square Root
One of the most common mistakes is overlooking the domain restriction imposed by the square root function. The square root of a real number is only defined for non-negative numbers. In our case, this means that sin(x) must be greater than or equal to zero. Failing to consider this restriction can lead to including values of x in the solution that make the square root undefined, rendering the entire expression meaningless. Remember, the domain restriction sin(x) ≥ 0 limits our focus to the intervals where the sine function is non-negative, which are the intervals [0, π] and their periodic repetitions. Ignoring this constraint can lead to a significantly incorrect solution set.
To avoid this mistake, always start by explicitly stating the domain restriction imposed by any square root functions in the inequality. This helps to keep the constraint at the forefront of your mind as you proceed with the solution. Before considering any other conditions, ensure that your final solution only includes values of x that satisfy the initial domain restriction. This is a fundamental step in solving inequalities involving square roots.
Forgetting the Strict Inequality
Another common mistake is not paying close attention to whether the inequality is strict (i.e., > or <) or non-strict (i.e., ≥ or ≤). In our case, we have a strict inequality: √sin(x) (sin(x) + 1/2) > 0. This means that the expression must be strictly greater than zero; it cannot be equal to zero. This distinction is crucial because it affects the endpoints of the solution intervals.
For example, when sin(x) = 0, the entire expression becomes zero, regardless of the value of sin(x) + 1/2. This occurs at integer multiples of π (i.e., x = nπ, where n is an integer). Therefore, these points must be excluded from the solution set. A common error is to include these points in the solution, resulting in an incorrect answer. Always carefully check the endpoints of your intervals to ensure they satisfy the strict inequality.
Incorrectly Solving Trigonometric Inequalities
Solving trigonometric inequalities like sin(x) > -1/2 can also be a source of errors if not handled carefully. It's important to understand the behavior of trigonometric functions on the unit circle and to use reference angles correctly. A common mistake is to only consider solutions within the interval [0, 2π) and forget to generalize the solution to all real numbers by adding integer multiples of 2π. Trigonometric functions are periodic, and their inequalities often have infinitely many solutions.
To solve trigonometric inequalities accurately, visualize the unit circle and identify the angles where the trigonometric function satisfies the inequality. Find the reference angles and use them to determine the intervals within one period. Then, generalize the solution by adding 2nπ (where n is an integer) to account for the periodic nature of the functions. This systematic approach helps to avoid overlooking solutions and ensures a complete answer.
Not Combining Conditions Properly
In complex inequalities, there are often multiple conditions that must be satisfied simultaneously. A common mistake is to solve each condition separately but fail to properly combine them to find the overall solution. For instance, in our example, we have the domain restriction sin(x) ≥ 0 and the condition sin(x) + 1/2 > 0. We need to find the intersection of the intervals that satisfy both conditions, not just the union.
To combine conditions correctly, it's helpful to represent the solution sets graphically or using interval notation. Identify the overlapping intervals, as these represent the values of x that satisfy all conditions. If there is no overlap, it means there is no solution to the inequality. Always double-check that your final solution satisfies all the individual conditions imposed by the inequality.
By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy in solving inequalities involving trigonometric functions and square roots. Careful attention to detail and a systematic approach are key to success in this area of mathematics.
Conclusion
In conclusion, solving the inequality √sin(x) (sin(x) + 1/2) > 0 requires a thorough understanding of the properties of square root and trigonometric functions. We began by analyzing the individual components of the inequality, focusing on the domain restrictions imposed by the square root and the behavior of the sine function. This led us to a step-by-step solution, where we first identified the domain restriction sin(x) ≥ 0, then solved the trigonometric inequality sin(x) > -1/2, and finally combined these conditions to arrive at the solution 2nπ < x < (2n + 1)π, where n is an integer. This solution represents the intervals where the given inequality holds true.
Throughout this exploration, we emphasized the importance of considering domain restrictions, understanding the strict inequality, correctly solving trigonometric inequalities, and properly combining different conditions. We also highlighted common mistakes and pitfalls that students often encounter, such as neglecting the domain restriction of the square root, forgetting the strict inequality, incorrectly solving trigonometric inequalities, and not combining conditions properly. By understanding these potential errors, we can approach similar problems with greater confidence and accuracy.
Furthermore, we discussed the value of graphical representation in visualizing the solution. Plotting the function y = √sin(x) (sin(x) + 1/2) allows us to see the intervals where the graph lies above the x-axis, providing a visual confirmation of our algebraic solution. This graphical approach enhances our understanding and provides a deeper insight into the behavior of the function and the inequality.
This comprehensive analysis demonstrates the multifaceted nature of mathematical problem-solving. It's not just about applying formulas; it's about understanding the underlying concepts, paying attention to detail, and connecting different ideas. By mastering these skills, we can tackle a wide range of mathematical challenges and deepen our appreciation for the beauty and precision of mathematics.