Finding The Exact Value Of Cosine 67.5 Degrees A Trigonometric Exploration
Hey guys! Let's dive into the fascinating world of trigonometry and figure out the exact value of $\cos(67.5^{\circ})$. This isn't your typical angle, so we'll need to use some cool trigonometric identities and half-angle formulas to crack this one. Buckle up, because we're about to embark on a mathematical adventure!
Understanding the Problem: Cosine 67.5 Degrees
When we're faced with finding the cosine of an angle like 67.5 degrees, which isn't a standard angle on the unit circle (like 0, 30, 45, 60, or 90 degrees), we need to get a little creative. This means we'll be reaching into our bag of trigonometric tricks to find a solution. The key here is to recognize that 67.5 degrees is exactly half of 135 degrees, which is a standard angle we know something about. This is where the half-angle formulas come into play. These formulas are our secret weapon for finding trigonometric functions of angles that are half of known angles. So, before we jump into the calculations, let's refresh our memory on what these half-angle formulas look like and how they can help us. We'll need the specific one for cosine, of course, but it's good to have the general idea in mind. Think of this as setting the stage for the main event – the actual calculation. We're building the foundation of our solution, making sure we have all the tools we need before we start the real work. Understanding the problem thoroughly is half the battle, and in this case, it leads us directly to the strategic use of half-angle formulas. This approach is not just about getting the right answer; it's about understanding why we're using a particular method. It's about connecting the dots between the problem, the tools, and the solution. So, let's keep this mindset as we move forward and unravel the mystery of $\cos(67.5^{\circ})$.
The Half-Angle Formula for Cosine
The half-angle formula for cosine is a powerful tool that allows us to find the cosine of an angle that is half of another angle, provided we know the cosine of the original angle. This formula is derived from the double-angle formula for cosine and is a staple in trigonometric problem-solving. The formula itself looks like this: $\cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}$. Notice the plus-or-minus sign in front of the square root. This is crucial because it tells us that we need to consider the quadrant in which the angle $\frac{\theta}{2}$ lies to determine the correct sign. In our case, we want to find $\cos(67.5^{\circ})$, and we know that 67.5 degrees is half of 135 degrees. So, we can set $\frac{\theta}{2} = 67.5^{\circ}$ and $\ heta = 135^{\circ}$. Now, we need to figure out the cosine of 135 degrees. Remember, 135 degrees is in the second quadrant, where cosine is negative. The reference angle for 135 degrees is 45 degrees (180 - 135 = 45), and we know that $\cos(45^{\circ}) = \frac{\sqrt{2}}{2}$. Therefore, $\cos(135^{\circ}) = -\frac{\sqrt{2}}{2}$. Now we have all the pieces we need to plug into the half-angle formula. We know $\cos(\theta)$, and we know the angle we're interested in, $\frac{\theta}{2}$. The next step is to substitute these values into the formula and simplify. But before we do that, let's pause and appreciate the elegance of this formula. It's a bridge that connects the cosine of an angle to the cosine of half that angle, allowing us to venture into angles that might otherwise seem out of reach. So, with our formula in hand and our values ready, let's move on to the calculation phase and see how this all comes together.
Applying the Formula: Calculation Steps
Alright, let's put the half-angle formula to work and calculate the exact value of $\cos(67.5^\circ})$. We've already established that $\cos(135^{\circ}) = -\frac{\sqrt{2}}{2}$, and we know our formula is $\cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}$. Now it's time to plug in the values. Substituting $\ heta = 135^{\circ}$ into the formula, we get) = \pm \sqrt\frac{1 + (-\frac{\sqrt{2}}{2})}{2}}$. The next step is to simplify the expression under the square root. First, let's get rid of the fraction within the fraction by multiplying both the numerator and the denominator inside the square root by 2) = \pm \sqrt\frac{2 - \sqrt{2}}{4}}$. Now we can take the square root of the denominator, which is 4, and pull it out of the square root) = \pm \frac\sqrt{2 - \sqrt{2}}}{2}}$. We're almost there! The final step is to determine the correct sign. Remember, 67.5 degrees lies in the first quadrant, where cosine is positive. Therefore, we choose the positive sign) = \frac{\sqrt{2 - \sqrt{2}}}{2}}$. And there you have it! We've successfully calculated the exact value of $\cos(67.5^{\circ})$ using the half-angle formula. This process demonstrates the power of trigonometric identities in solving problems that might seem daunting at first. By breaking down the problem into smaller steps and applying the appropriate formulas, we can tackle even the trickiest trigonometric challenges. So, let's take a moment to appreciate the journey we've been on, from understanding the problem to applying the formula and arriving at the solution. It's not just about the answer; it's about the process and the understanding we've gained along the way.
The Final Answer and its Significance
So, after all that mathematical maneuvering, we've arrived at our final answer: $\cos(67.5^{\circ}) = \frac{\sqrt{2 - \sqrt{2}}}{2}}$. This is the exact value, and it's pretty cool, right? It's not something you'd easily guess, and it highlights the beauty and precision of trigonometry. But what does this answer really mean? Well, it tells us the x-coordinate of the point on the unit circle that corresponds to an angle of 67.5 degrees. It's a specific, precise value, not just an approximation from a calculator. This exactness is crucial in many areas of mathematics and physics, where approximations can lead to inaccuracies in more complex calculations. Think about fields like engineering, where precise measurements and calculations are essential for designing structures or systems. Or consider computer graphics, where trigonometric functions are used to create realistic images and animations. Having exact values allows for greater accuracy and control in these applications. Moreover, finding this value wasn't just about plugging into a formula. It was about understanding the relationship between angles, their trigonometric functions, and the underlying principles of trigonometry. We used the half-angle formula, which is a direct consequence of other fundamental trigonometric identities. This highlights the interconnectedness of mathematical concepts and the power of building upon foundational knowledge. So, while the answer itself is important, the process we went through to get there is equally valuable. It's about developing problem-solving skills, understanding mathematical relationships, and appreciating the elegance of exact solutions. And who knows, maybe this journey has sparked a deeper interest in trigonometry and its applications. That's the real significance of finding this answer – it's a step on the path to mathematical understanding and exploration.
Why Option B is the Correct Answer
Now that we've gone through the process of calculating the exact value of $\cos(67.5^{\circ})$, let's connect our result to the multiple-choice options presented in the question. We found that $\cos(67.5^{\circ}) = \frac{\sqrt{2 - \sqrt{2}}}{2}}$. Looking at the options, we can see that this matches option B. The other options are either negative or have a different structure within the square root, so they don't align with our calculated value. Option A, $-\frac{\sqrt{2+\sqrt{2}}}{2}$, is negative, and we know that cosine is positive in the first quadrant where 67.5 degrees lies. So, we can eliminate this option. Option C, $-\frac{\sqrt{2+\sqrt{2}}}{4}$, is also negative and has a different denominator, making it incorrect as well. Therefore, by carefully working through the calculation and understanding the properties of cosine in different quadrants, we can confidently say that option B is the correct answer. This process of elimination is a valuable strategy in multiple-choice questions. By understanding why certain options are incorrect, we can narrow down our choices and increase our confidence in the correct answer. It's not just about finding the right answer; it's about understanding why the other answers are wrong. This deeper level of understanding is what truly solidifies our knowledge and allows us to apply it in different contexts. So, congratulations on correctly identifying option B as the answer! You've demonstrated a strong grasp of trigonometric principles and problem-solving techniques.
Conclusion: The Power of Trigonometric Identities
In conclusion, finding the exact value of $\cos(67.5^{\circ})$ is a testament to the power and elegance of trigonometric identities. We embarked on this journey by recognizing that 67.5 degrees is half of 135 degrees, which led us to the half-angle formula for cosine. This formula became our key to unlocking the solution. We carefully applied the formula, simplified the expression, and considered the quadrant in which 67.5 degrees lies to determine the correct sign. Through this process, we not only found the exact value, which is $\frac{\sqrt{2 - \sqrt{2}}}{2}}$, but also reinforced our understanding of trigonometric principles and problem-solving techniques. We saw how seemingly complex problems can be broken down into smaller, manageable steps, and how the relationships between different trigonometric functions can be leveraged to our advantage. This exercise highlights the importance of mastering fundamental concepts and formulas in trigonometry. The half-angle formula is just one tool in a vast toolbox, but it's a powerful one when applied correctly. It allows us to venture beyond the standard angles on the unit circle and find exact values for a wider range of angles. Moreover, this journey reminds us that mathematics is not just about finding the right answer; it's about the process of discovery and the understanding we gain along the way. It's about connecting the dots between different concepts and appreciating the beauty and precision of mathematical solutions. So, let's continue to explore the world of trigonometry, embrace the challenges, and celebrate the power of mathematical thinking. Who knows what other fascinating values and relationships we'll uncover along the way? Keep exploring, keep questioning, and keep learning!