Finding Sin(22.5°) Using The Half-Angle Formula A Step-by-Step Solution

In the realm of trigonometry, half-angle formulas serve as powerful tools for determining the trigonometric function values of angles that are half the size of known angles. This article delves into the application of the half-angle formula to calculate the sine of 22.5 degrees, a seemingly complex value that can be elegantly derived using this method. We will embark on a step-by-step journey, unraveling the formula, applying it to our specific case, simplifying the result, and expressing it in fractional form. This exploration will not only provide a solution to the problem but also enhance your understanding of trigonometric identities and their practical applications. Trigonometric identities are fundamental to solving a wide array of problems in mathematics, physics, and engineering. By mastering these concepts, you gain a deeper appreciation for the interconnectedness of mathematical ideas and their relevance to the world around us. This article will provide a clear and concise explanation of the process, ensuring that readers of all backgrounds can follow along and grasp the underlying principles. We will also emphasize the importance of simplifying the final answer and expressing it in the correct form, a crucial aspect of mathematical rigor. So, let's embark on this journey together and unlock the secrets of the half-angle formula!

Understanding the Half-Angle Formula for Sine

The half-angle formula for sine is a cornerstone of trigonometric identities, providing a direct relationship between the sine of an angle and the cosine of its double. This formula is particularly useful when we need to find the sine of an angle that is half of a well-known angle, such as those found on the unit circle (0°, 30°, 45°, 60°, 90°, etc.). The formula itself is expressed as follows:

$$\sin(\frac{x}{2}) = \pm \sqrt{\frac{1 - \cos(x)}{2}}$$

Where:

• sin(x/2) represents the sine of half the angle x.
• cos(x) represents the cosine of the angle x.
• The ± symbol indicates that the result can be either positive or negative, depending on the quadrant in which the angle x/2 lies. This is a crucial point to remember, as the sign of the sine function varies across different quadrants. In the first and second quadrants, sine is positive, while in the third and fourth quadrants, it is negative. To determine the correct sign, we need to consider the angle x/2 and identify its quadrant. The formula stems from the double-angle formula for cosine, which can be rearranged and manipulated to isolate the sine term. This connection highlights the interconnectedness of trigonometric identities and their ability to be derived from one another. The half-angle formula is not just a standalone equation; it's part of a larger family of trigonometric relationships that enable us to solve a wide variety of problems. Understanding the origin and derivation of the formula provides a deeper appreciation for its utility and application. The $\pm$ symbol in the formula is a critical aspect that must be carefully considered when applying the formula. For instance, if we are finding the sine of an angle in the first quadrant, we would choose the positive square root, as sine is positive in the first quadrant. Conversely, if the angle lies in the third quadrant, we would select the negative square root. This quadrantal awareness is essential for accurate calculations. The half-angle formula is a powerful tool, but it's important to use it judiciously and with a clear understanding of its nuances. By mastering this formula and its applications, you can significantly expand your ability to solve trigonometric problems.

Applying the Formula to Find sin(22.5°)

Now, let's put the half-angle formula into action and calculate sin(22.5°). We can recognize that 22.5° is exactly half of 45°, an angle whose trigonometric values we know precisely. This makes the half-angle formula perfectly suited for this problem. Our goal is to find sin(22.5°), which can be written as sin(45°/2). This means we will use x = 45° in our half-angle formula. The cosine of 45° is a well-known value: cos(45°) = √2 / 2. This value is derived from the unit circle or the properties of a 45-45-90 triangle, where the two legs are equal in length, and the hypotenuse is √2 times the length of a leg. Substituting this value into the half-angle formula, we get:

$$\sin(22.5^\circ) = \pm \sqrt{\frac{1 - \cos(45^\circ)}{2}} = \pm \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}$$

Before we proceed further, we need to determine the correct sign. Since 22.5° lies in the first quadrant, where sine is positive, we will take the positive square root. This is a crucial step, as choosing the wrong sign would lead to an incorrect answer. The angle 22.5° is a relatively small angle, and visualizing it on the unit circle confirms that its sine value should be positive. Now, we can focus on simplifying the expression inside the square root. We have a fraction within a fraction, which can be simplified by finding a common denominator. The common denominator for 1 and √2 / 2 is 2. So, we can rewrite 1 as 2/2. This allows us to combine the terms in the numerator.

$$\sin(22.5^\circ) = \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}}$$

Next, we can simplify the complex fraction by dividing by 2, which is the same as multiplying by 1/2. This will combine the denominators and lead us closer to our final simplified answer. This step-by-step approach ensures that we handle each operation carefully and minimize the risk of errors. The process of applying the half-angle formula involves not only substituting the correct values but also making strategic decisions about simplification and sign selection. By understanding the underlying principles and applying them methodically, we can successfully navigate these challenges.

Simplifying the Expression

Now, let's focus on simplifying the expression we obtained in the previous section. We have:

$$\sin(22.5^\circ) = \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}}$$

To simplify this complex fraction, we can multiply the numerator and denominator of the main fraction by 1/2, which is equivalent to dividing by 2:

$$\sin(22.5^\circ) = \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}} = \sqrt{\frac{2 - \sqrt{2}}{4}}$$

We now have a single fraction under the square root. To further simplify, we can take the square root of the denominator, which is √4 = 2:

$$\sin(22.5^\circ) = \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2}$$

This expression is significantly simpler than our initial result, but we can still explore further simplification. The presence of a square root within a square root suggests that we might be able to rationalize the numerator. However, in this case, rationalizing the numerator doesn't lead to a simpler form. The expression $\sqrt{2 - \sqrt{2}}$ is already in its simplest radical form. Therefore, we have arrived at the most simplified form of sin(22.5°). This process of simplification highlights the importance of recognizing when an expression is in its simplest form. While further algebraic manipulations might be possible, they might not necessarily lead to a more elegant or useful result. The ability to identify the simplest form is a key skill in mathematics. The final simplified expression, $\frac{\sqrt{2 - \sqrt{2}}}{2}$, is an exact value for sin(22.5°). This is in contrast to decimal approximations, which would introduce rounding errors. By expressing the result in this form, we maintain the precision of the solution. The process of simplifying radical expressions often involves techniques such as rationalizing the denominator, factoring out perfect squares, and combining like terms. In this case, we employed the strategy of simplifying a complex fraction and then evaluating the square root of the denominator. Each step in the simplification process was guided by the goal of obtaining the most concise and transparent representation of the solution. The final result, $\frac{\sqrt{2 - \sqrt{2}}}{2}$, showcases the beauty and elegance of mathematical expressions. It demonstrates how seemingly complex values can be expressed in a precise and compact form.

Expressing the Answer in Fractional Form

Our simplified answer, $\frac{\sqrt{2 - \sqrt{2}}}{2}$, is already expressed in fractional form. The key characteristic of a fractional form is that it represents a number as a ratio of two integers. In our case, the denominator is the integer 2, and the numerator is the radical expression $\sqrt{2 - \sqrt{2}}$. While the numerator is not an integer, the entire expression is still considered to be in fractional form because it represents a fraction where the denominator is an integer. The term "fractional form" in mathematics often implies that the denominator is a whole number. In this context, we have successfully expressed sin(22.5°) as a fraction with a denominator of 2. This form is both exact and simplified, fulfilling the requirements of the problem. It is important to distinguish between fractional form and rational form. A rational number can be expressed as a fraction p/q, where p and q are both integers and q is not zero. Our answer, $\frac{\sqrt{2 - \sqrt{2}}}{2}$, is in fractional form but not rational form because the numerator is not an integer. The instruction to express the answer in fractional form was specifically intended to guide us towards this type of representation. If the instruction had been to express the answer in rational form, we would have needed to explore additional simplification techniques, which might not have been possible in this case. The fractional form of the answer provides a clear and concise representation of the value of sin(22.5°). It allows us to easily compare this value with other trigonometric values and to use it in further calculations. The fractional form also highlights the relationship between sin(22.5°) and the square root of a difference involving √2. This connection can be further explored using other trigonometric identities and formulas. The process of expressing the answer in fractional form emphasizes the importance of understanding mathematical terminology and the nuances of different types of representations. By carefully adhering to the instructions and applying the appropriate mathematical principles, we can arrive at the correct and most suitable form of the solution.

In conclusion, we have successfully utilized the half-angle formula to find the exact value of sin(22.5°). We began by understanding the formula itself, recognizing its connection to the double-angle formula for cosine, and emphasizing the importance of considering the sign based on the quadrant of the angle. We then applied the formula to our specific case, substituting cos(45°) and carefully simplifying the resulting expression. Through a series of algebraic manipulations, we arrived at the simplified form: $\frac{\sqrt{2 - \sqrt{2}}}{2}$. This final answer is not only simplified but also expressed in the requested fractional form. The journey of solving this problem has provided valuable insights into the application of trigonometric identities, the importance of simplification techniques, and the nuances of mathematical representation. The half-angle formula is a powerful tool in trigonometry, allowing us to find the trigonometric function values of angles that are half of known angles. By mastering this formula and its applications, you can expand your ability to solve a wide range of trigonometric problems. The process of simplification is a crucial aspect of mathematical problem-solving. It involves applying algebraic techniques to reduce an expression to its most concise and transparent form. In this case, we simplified a complex fraction and extracted the square root of the denominator to arrive at our final answer. The final answer, $\frac{\sqrt{2 - \sqrt{2}}}{2}$, represents the exact value of sin(22.5°). This exact value is preferable to decimal approximations, which would introduce rounding errors. By expressing the answer in this form, we maintain the precision of the solution. The ability to express mathematical results in different forms is an important skill. In this case, we were specifically asked to express the answer in fractional form. By understanding the meaning of "fractional form" and applying the appropriate techniques, we were able to successfully meet this requirement. This exploration of finding sin(22.5°) using the half-angle formula serves as a testament to the elegance and power of trigonometric identities. It demonstrates how seemingly complex values can be derived and expressed in a precise and meaningful way. By continuing to explore these concepts and practice their application, you can deepen your understanding of trigonometry and its role in mathematics and beyond.