Simplifying $2 \cos^2(157.5^{\circ}) - 1$: A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little trigonometric problem: simplifying the expression $2 \cos^2(157.5^{\circ}) - 1$. This might look a bit intimidating at first glance, but don't worry, we'll break it down step by step and make it super easy to understand. We will leverage trigonometric identities, especially the double-angle formula, to simplify this expression. So, let's get started and unlock the secrets hidden within this equation!

Understanding the Problem

Before we jump into solving, let's make sure we understand what we're dealing with. The expression $2 \cos^2(157.5^{\circ}) - 1$ involves a cosine function raised to the power of 2, and a specific angle, 157. 5 degrees. Our goal is to simplify this expression, ideally to a single numerical value or a simpler trigonometric form. This involves leveraging trigonometric identities, which are equations that are always true for any value of the angles. These identities act as our tools in simplifying complex expressions into more manageable forms. Trigonometric identities are the backbone of simplifying trigonometric expressions. The expression clearly hints at the use of the double-angle formula, specifically for cosine. Recognizing these patterns early on is crucial for efficient problem-solving. Simplifying this expression can help us better understand the relationships between different trigonometric functions and how they behave at various angles. Now, let's dive into the mathematical tools we'll need to tackle this problem.

Key Trigonometric Identity: The Double-Angle Formula

The key to simplifying this expression lies in recognizing a very handy trigonometric identity: the double-angle formula for cosine. There are a few forms of this formula, but the one that's most relevant to our problem is: $\cos(2\theta) = 2\cos^2(\theta) - 1$. Notice anything familiar? Our expression, $2 \cos^2(157.5^{\circ}) - 1$, looks exactly like the right side of this identity! This is a major clue and a big help. This identity essentially tells us that if we have an expression in the form of $2\cos^2(\theta) - 1$, we can directly replace it with $\cos(2\theta)$. It's like having a magic key that unlocks a simpler form. The double-angle formula is a cornerstone of trigonometry, allowing us to relate trigonometric functions of an angle to those of double that angle. Mastering these identities not only simplifies calculations but also deepens our understanding of trigonometric relationships. By recognizing and applying the double-angle formula, we can transform a seemingly complex expression into a much more manageable one. This is a powerful technique that's widely used in various fields, including physics, engineering, and computer graphics. This identity simplifies the problem significantly, allowing us to bypass complicated calculations. Now that we have the right tool, let's apply it to our problem!

Applying the Identity

Okay, so we know that $\cos(2\theta) = 2\cos^2(\theta) - 1$. In our case, $\theta$ is 157.5 degrees. Let's substitute that into the identity: $\cos(2 * 157.5^{\circ}) = 2\cos^2(157.5^{\circ}) - 1$. Now, we just need to calculate 2 * 157.5 degrees, which equals 315 degrees. So, our expression simplifies to $\cos(315^{\circ})$. See how much simpler that is? We've gone from a somewhat complex expression to a single cosine value. By directly substituting the given angle into the double-angle formula, we've effectively reduced the problem to evaluating the cosine of a single angle. This step highlights the elegance and efficiency of using trigonometric identities. It transforms a problem that might have seemed daunting into a straightforward calculation. Understanding how to substitute values correctly into identities is a critical skill in trigonometry. It allows us to manipulate expressions and solve problems with greater ease and accuracy. The simplification achieved here underscores the power of trigonometric identities in streamlining calculations. Now that we've simplified our expression to a single cosine value, let's move on to evaluating it.

Evaluating $\cos(315^{\circ})$

Now, we need to find the value of $\cos(315^{\circ})$. To do this, let's think about the unit circle. Remember, the unit circle 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. 315 degrees is in the fourth quadrant. In the fourth quadrant, cosine is positive. To find the reference angle (the acute angle formed between the terminal side of our angle and the x-axis), we subtract 315 from 360: $360^{\circ} - 315^{\circ} = 45^{\circ}$. So, $\cos(315^{\circ})$ is the same as $\cos(45^{\circ})$. And we know that $\cos(45^{\circ}) = \frac{\sqrt{2}}{2}$. Therefore, $\cos(315^{\circ}) = \frac{\sqrt{2}}{2}$. Understanding the unit circle is fundamental to evaluating trigonometric functions. It provides a visual representation of how angles and their trigonometric values relate. By visualizing 315 degrees on the unit circle, we can quickly determine its quadrant and the sign of its cosine. The concept of a reference angle simplifies the evaluation process, allowing us to relate the trigonometric values of angles in different quadrants to those of acute angles. Remembering common trigonometric values, such as $\cos(45^{\circ}) = \frac{\sqrt{2}}{2}$, is essential for efficient problem-solving. This step demonstrates the importance of both conceptual understanding and memorization in trigonometry. Now that we've evaluated the cosine value, we have our final answer!

Final Answer

So, after all that, we've simplified $2 \cos^2(157.5^{\circ}) - 1$ to $\cos(315^{\circ})$, and then found that $\cos(315^{\circ}) = \frac{\sqrt{2}}{2}$. Therefore, $2 \cos^2(157.5^{\circ}) - 1 = \frac{\sqrt{2}}{2}$. Awesome! We did it! This final answer represents the simplified form of our original expression. It's a single numerical value that encapsulates the result of all our calculations. This result underscores the effectiveness of using trigonometric identities and the unit circle to simplify and evaluate trigonometric expressions. By breaking down the problem into smaller, manageable steps, we were able to arrive at a clear and concise solution. This process highlights the beauty of mathematics in transforming complex problems into elegant solutions. This final answer not only provides the solution but also reinforces the importance of each step taken to achieve it.

Key Takeaways

Let's recap what we learned today:

• We used the double-angle formula for cosine: $\cos(2\theta) = 2\cos^2(\theta) - 1$.
• We applied this identity to simplify the expression $2 \cos^2(157.5^{\circ}) - 1$ to $\cos(315^{\circ})$.
• We used the unit circle to find that $\cos(315^{\circ}) = \frac{\sqrt{2}}{2}$.

Mastering these concepts will be super helpful in your future math adventures. Remember, practice makes perfect, so keep tackling those trig problems! These key takeaways provide a concise summary of the main concepts and techniques used in solving the problem. They serve as a valuable tool for reinforcing learning and facilitating future problem-solving. Understanding and remembering these key points will enable you to tackle similar trigonometric problems with confidence. The ability to recall and apply these concepts is crucial for building a strong foundation in trigonometry. By focusing on the key takeaways, you can effectively consolidate your knowledge and enhance your problem-solving skills. So keep practicing and you'll get even better at trigonometry!

Conclusion

Simplifying trigonometric expressions can seem tricky, but by using the right identities and understanding the unit circle, you can conquer even the most intimidating problems. We hope this step-by-step guide helped you understand how to simplify $2 \cos^2(157.5^{\circ}) - 1$. Keep practicing, and you'll become a trig whiz in no time! Remember, the key to mastering trigonometry is consistent practice and a solid understanding of fundamental concepts. By breaking down complex problems into smaller, manageable steps and applying the appropriate trigonometric identities, you can achieve elegant solutions. This process not only enhances your problem-solving skills but also deepens your appreciation for the beauty and interconnectedness of mathematics. So, keep exploring, keep practicing, and keep pushing your boundaries in the world of trigonometry! You've got this! If you have any questions or want to explore more trig problems, feel free to ask! Happy simplifying!