Cos 22.5° Exact Value: Half-Angle Identity Explained

by Rajiv Sharma 53 views

Hey guys! Today, we're diving into the exciting world of trigonometry, specifically focusing on how to find the exact value of trigonometric functions using half-angle identities. These identities are super handy when dealing with angles that are half of a standard angle whose trigonometric values we already know. In this article, we'll break down the process step-by-step, making it easy to understand and apply. We're going to tackle the question: What is the exact value of $\cos 22.5^{\circ}$? So, grab your calculators (though you won't need them for the final answer!), and let's get started!

The half-angle identities are a powerful tool in trigonometry, allowing us to determine the trigonometric functions of angles that are half the measure of angles we already know. These identities are derived from the double-angle formulas and provide a straightforward method for calculations that would otherwise be quite complex. Understanding and applying these identities can greatly simplify problems involving trigonometric functions, especially when dealing with non-standard angles. The half-angle identities for cosine, sine, and tangent are as follows:

cos(x2)=pmsqrtfrac1+cos(x)2\\cos(\frac{x}{2}) = \\pm \\sqrt{\\frac{1 + \\cos(x)}{2}}

sin(x2)=pmsqrtfrac1−cos(x)2sin(\frac{x}{2}) = \\pm \\sqrt{\\frac{1 - \\cos(x)}{2}}

tan(x2)=pmsqrtfrac1−cos(x)1+cos(x)=fracsin(x)1+cos(x)=frac1−cos(x)sin(x)tan(\frac{x}{2}) = \\pm \\sqrt{\\frac{1 - \\cos(x)}{1 + \\cos(x)}} = \\frac{sin(x)}{1 + cos(x)} = \\frac{1 - cos(x)}{sin(x)}

These formulas provide a direct way to calculate the trigonometric function of a half-angle based on the cosine of the full angle. The plus-minus sign (\$\pm\$) indicates that we need to consider the quadrant in which the half-angle lies to determine the correct sign. Remember, the quadrant dictates whether the trigonometric function is positive or negative. The choice of sign depends on which quadrant the angle x/2x/2 falls into. For example, if x/2x/2 is in the first or fourth quadrant, cos(x/2)\\cos(x/2) will be positive. If x/2x/2 is in the second or third quadrant, cos(x/2)\\cos(x/2) will be negative. Similarly, we consider the quadrant for sine and tangent to determine the appropriate sign. These identities are derived from the double-angle formulas and the Pythagorean identity, making them a cornerstone of trigonometric calculations.

Before we jump into our specific problem, let's quickly recap what half-angle identities are all about. Basically, they're formulas that allow us to find the trigonometric functions (like sine, cosine, and tangent) of an angle that's half the size of another angle. These identities are particularly useful when we need to find the exact values of trigonometric functions for angles that aren't the usual suspects (like 30°, 45°, 60°, etc.). The half-angle identities are derived from the double-angle formulas, and they look like this:

  • ${\cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}\}$
  • ${sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}\}$
  • ${tan(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}\}$

The symbol $\pm$ might look a bit confusing, but don't worry! It just means we need to choose either the positive or negative square root, depending on which quadrant the angle $\frac{\theta}{2}$ lies in. The beauty of half-angle identities lies in their ability to simplify complex trigonometric calculations into more manageable algebraic expressions. By utilizing these identities, we can evaluate trigonometric functions for a wide range of angles, including those that are not part of the standard trigonometric table. This is especially useful in fields such as physics and engineering, where trigonometric functions are frequently used to model various phenomena. The choice between the positive and negative sign is determined by the quadrant of the angle $\theta/2$, which is crucial for obtaining the correct result. The derivation of these identities involves algebraic manipulation and the use of other trigonometric identities, such as the Pythagorean identity and the double-angle formulas. This makes them a central concept in trigonometry, bridging the gap between basic trigonometric functions and more advanced applications. Understanding and mastering half-angle identities is essential for anyone looking to deepen their knowledge of trigonometry and its applications.

Okay, let's get to the heart of the matter: finding the exact value of $\cos 22.5^{\circ}$. The trick here is to recognize that 22.5° is half of 45°, and we know the exact value of $\cos 45^{\circ}$. That's where the half-angle identity for cosine comes to the rescue! Using the formula, we have:

cos(theta2)=pmsqrtfrac1+cos(theta)2\\cos(\frac{\\theta}{2}) = \\pm \\sqrt{\\frac{1 + \\cos(\\theta)}{2}}

In our case, $\theta = 45^{\circ}$,so$fractheta2=22.5circ\$, so \$\\frac{\\theta}{2} = 22.5^{\\circ}$. Plugging this into the half-angle identity, we get:

cos(22.5circ)=pmsqrtfrac1+cos(45circ)2\\cos(22.5^{\\circ}) = \\pm \\sqrt{\\frac{1 + \\cos(45^{\\circ})}{2}}

Now, we know that $\cos(45^{\circ}) = \frac{\sqrt{2}}{2}$, so we can substitute that in:

cos(22.5circ)=pmsqrtfrac1+fracsqrt222\\cos(22.5^{\\circ}) = \\pm \\sqrt{\\frac{1 + \\frac{\\sqrt{2}}{2}}{2}}

Before we proceed further, it's important to discuss the sign. Since 22.5° is in the first quadrant, where cosine is positive, we'll take the positive square root. This is a critical step in using the half-angle identity because the $\pm$ symbol indicates that we need to consider the quadrant of the resulting angle to determine the correct sign. In the first quadrant, all trigonometric functions are positive, so we know that $\cos(22.5^{\circ})$ will be positive. The choice of sign is not just a formality; it directly affects the accuracy of the final result. Ignoring the quadrant can lead to a completely incorrect answer. The consideration of the sign is a testament to the geometric interpretation of trigonometric functions and their behavior in different quadrants. It reinforces the idea that these functions are not just abstract mathematical entities but are closely tied to the unit circle and the angles formed within it. Therefore, when applying half-angle identities, always take a moment to determine the correct sign based on the quadrant of the angle. This careful attention to detail ensures that the solution is both mathematically sound and geometrically consistent, leading to a reliable and accurate result. By correctly applying the sign, we maintain the integrity of the trigonometric relationships and avoid any potential pitfalls in the calculation process.

Alright, let's simplify that expression we landed on:

cos(22.5circ)=sqrtfrac1+fracsqrt222\\cos(22.5^{\\circ}) = \\sqrt{\\frac{1 + \\frac{\\sqrt{2}}{2}}{2}}

To make things easier, let's get rid of the fraction within the fraction. We can do this by multiplying both the numerator and the denominator inside the square root by 2:

cos(22.5circ)=sqrtfrac2+sqrt24\\cos(22.5^{\\circ}) = \\sqrt{\\frac{2 + \\sqrt{2}}{4}}

Now, we can take the square root of the denominator, which is 4:

cos(22.5circ)=fracsqrt2+sqrt22\\cos(22.5^{\\circ}) = \\frac{\\sqrt{2 + \\sqrt{2}}}{2}

And there you have it! This is the exact value of $\cos 22.5^{\circ}$. This simplification process is crucial in obtaining a clean and understandable final answer. The initial expression, while mathematically correct, is not in its simplest form and can be cumbersome to work with in further calculations. By eliminating the fraction within the fraction, we make the expression more manageable and easier to interpret. The step-by-step simplification not only makes the final answer more aesthetically pleasing but also reduces the chances of making errors in subsequent calculations. The ability to simplify complex expressions is a fundamental skill in mathematics, and this example demonstrates how it can be applied in the context of trigonometric functions. Each step in the simplification process is deliberate and aims to transform the expression into a more concise and transparent form. The final form, $\frac{\sqrt{2 + \sqrt{2}}}{2}$, clearly represents the exact value of $\cos 22.5^{\circ}$ and is easily recognizable as a standard trigonometric result. The process of rationalizing the denominator and simplifying radicals is a key aspect of obtaining the final answer in its most refined form. This ensures that the answer is not only mathematically accurate but also presented in a way that is easily understood and can be readily used in other mathematical contexts. The simplification highlights the elegance and precision of mathematical operations, where each step contributes to the overall clarity and accuracy of the solution.

So, the exact value of $\cos 22.5^{\circ}$ is:

cos(22.5circ)=fracsqrt2+sqrt22\\cos(22.5^{\\circ}) = \\frac{\\sqrt{2 + \\sqrt{2}}}{2}

This is our final answer. Isn't it neat how we could find this exact value using the half-angle identity? No decimal approximations here – just a beautiful, precise answer! Understanding how to use these identities opens up a whole new world of trigonometric calculations, allowing us to find the exact values for many more angles. The ability to express trigonometric values in exact form is crucial in various fields of mathematics and physics, where precision is paramount. Decimal approximations, while convenient for some applications, can introduce rounding errors that accumulate over multiple calculations. Exact values, on the other hand, provide a level of accuracy that is essential in theoretical work and certain practical applications. This result also highlights the interconnectedness of different areas within trigonometry. The half-angle identity itself is derived from the double-angle formulas, which in turn are related to the angle sum and difference identities. This web of relationships underscores the importance of having a solid foundation in trigonometric principles. The final answer, $\frac{\sqrt{2 + \sqrt{2}}}{2}$, is a testament to the power of these identities and their ability to unlock the exact values of trigonometric functions for a wide range of angles. It serves as a reminder that mathematics is not just about memorizing formulas but also about understanding the underlying principles and how they can be applied to solve problems creatively and precisely. The ability to manipulate and simplify trigonometric expressions is a valuable skill that extends beyond the classroom and into real-world applications in science, engineering, and other fields.

In this article, we've successfully found the exact value of $\cos 22.5^{\circ}$ using the half-angle identity. We walked through the steps, from understanding the identity to simplifying the final expression. Hopefully, you guys now feel more confident in tackling similar problems. Remember, practice makes perfect, so keep those trigonometric wheels turning! Mastering the use of half-angle identities is a significant step in developing a comprehensive understanding of trigonometry. These identities are not just isolated formulas but are part of a larger framework of trigonometric relationships and concepts. By understanding how to apply them effectively, you can tackle a wide range of trigonometric problems with confidence and precision. The ability to find exact values, as demonstrated in this example, is particularly valuable in many scientific and engineering contexts. The process of working through a problem like this also reinforces the importance of attention to detail and careful algebraic manipulation. Each step, from substituting the known values to simplifying the expression, requires a clear understanding of the underlying principles and a commitment to accuracy. Trigonometry is a fundamental branch of mathematics with applications in various fields, including physics, engineering, computer graphics, and navigation. A strong grasp of trigonometric identities and their applications is essential for success in these areas. The half-angle identities, in particular, are useful tools for solving problems involving angles that are not standard angles. By understanding and practicing their use, you can expand your mathematical toolkit and enhance your problem-solving abilities. The journey through trigonometry is one of discovery and exploration, and the half-angle identities are just one of the many fascinating aspects of this field. Keep exploring, keep practicing, and you'll continue to unlock new mathematical insights and capabilities.