Calculating Sin(7π/12) * Sin(11π/12): A Step-by-Step Guide
Hey everyone! Today, we're diving into a fun little trigonometry problem: figuring out the value of sin(7π/12) * sin(11π/12). Don't worry if it looks a bit intimidating at first; we'll break it down step by step, making it super easy to follow. This is a classic example of how we can use trigonometric identities to simplify seemingly complex expressions. So, grab your calculators (or your brains!) and let's get started. We'll explore the problem in detail and provide a clear, concise solution. This problem is very important, because it allows us to show that even complex problems can be simplified by applying the right trigonometric identities and a little bit of algebraic manipulation. We'll be using the sum and difference formulas for sine, the double-angle formulas, and a couple of other handy tricks to get to our final answer. The key is to recognize that we can rewrite the angles 7π/12 and 11π/12 in terms of angles whose sine and cosine values we already know, like π/6, π/4, and π/3. Let's start with a refresher on the basics, and then we'll jump right into the heart of the problem. Also, We'll provide plenty of context and explanations so that you don't feel lost, and the whole process will become clear. So, let's get started. We aim to clarify the steps involved and provide a comprehensive understanding of the solution. By the end of this guide, you'll be able to confidently solve similar problems. This method can be applied to a variety of other problems involving trigonometric functions. This approach ensures that we don't just get the answer but also understand why the answer is what it is. It's all about mastering the concepts, not just memorizing the formulas.
Breaking Down the Problem: Understanding the Basics
Alright, before we get our hands dirty with the calculations, let's quickly recap some fundamental trigonometric concepts. Sine is a trigonometric function that relates an angle of a right-angled triangle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. The values of sine for some common angles like 0, π/6, π/4, π/3, and π/2 (or 0°, 30°, 45°, 60°, and 90° in degrees) are something you should ideally have at your fingertips. For example, sin(π/6) = 1/2, sin(π/4) = √2/2, and sin(π/3) = √3/2. You'll encounter these values frequently, so knowing them will save you a lot of time. Also, remember that the sine function is positive in the first and second quadrants and negative in the third and fourth quadrants of the unit circle. This is important for determining the sign of your final answer. Another key concept is the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate system. Each point on the unit circle corresponds to an angle and its sine and cosine values. The sine of an angle is the y-coordinate of the point where the terminal side of the angle intersects the unit circle. Furthermore, we must understand trigonometric identities. These are equations that are true for all values of the variables involved. We will use them to simplify the expression and to solve the problem. Now that we have covered the basic concepts, we are ready to move on and solve the problem. These identities will be our main tools for solving this problem. They allow us to rewrite the expression in a way that is easier to work with. These concepts are the foundation of trigonometry, and understanding them is crucial for solving problems like these. The core idea is that we can manipulate trigonometric expressions to make them simpler. Remember, practice is key! The more you work with these concepts, the more comfortable you'll become. Each step builds upon the previous one, so make sure you understand each part before moving on.
Step-by-Step Solution: Unraveling sin(7π/12) * sin(11π/12)
Okay, time to get to the main event! We want to find the value of sin(7π/12) * sin(11π/12). The first thing we can do is rewrite the angles 7π/12 and 11π/12 as sums or differences of angles whose sine and cosine values we know. Notice that:
7π/12 = (3π/12) + (4π/12) = π/4 + π/311π/12 = (8π/12) + (3π/12) = 2π/3 + π/4
Now we can use the sum and difference formulas for sine. Recall that sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Applying this formula to sin(7π/12), we get:
sin(7π/12) = sin(π/4 + π/3) = sin(π/4)cos(π/3) + cos(π/4)sin(π/3)
Using the known values for sine and cosine of π/4 and π/3:
sin(7π/12) = (√2/2)(1/2) + (√2/2)(√3/2) = √2/4 + √6/4 = (√2 + √6)/4
Similarly, we can find sin(11π/12):
sin(11π/12) = sin(2π/3 + π/4) = sin(2π/3)cos(π/4) + cos(2π/3)sin(π/4)
We know sin(2π/3) = √3/2 and cos(2π/3) = -1/2, so:
sin(11π/12) = (√3/2)(√2/2) + (-1/2)(√2/2) = √6/4 - √2/4 = (√6 - √2)/4
Now we have sin(7π/12) and sin(11π/12). Let's multiply them together:
sin(7π/12) * sin(11π/12) = [(√2 + √6)/4] * [(√6 - √2)/4]
This is where we use the difference of squares: (a + b)(a - b) = a² - b²:
sin(7π/12) * sin(11π/12) = [(√6)² - (√2)²] / 16 = (6 - 2) / 16 = 4/16 = 1/4
Therefore, the final answer is 1/4. We have shown all the steps needed to solve the problem. We used basic trigonometric identities and algebraic manipulations to arrive at our answer. We simplified the expression using well-known identities and calculated the values of sine for the given angles. This methodical approach ensures that you grasp the concepts and can apply them to other problems. The use of known trigonometric values and algebraic manipulation is crucial. We started with the basic definitions and then proceeded to use the sum and difference formulas for sine. The final step was to simplify the resulting expression to get the answer. We have arrived at the final answer by breaking down the problem into smaller, manageable steps.
Alternative Approach: Using Product-to-Sum Identities
Alright, guys, let's explore an alternative approach to solve this problem. This method leverages product-to-sum identities. These identities allow us to rewrite the product of sines (or cosines) as a sum or difference of cosines (or sines). The relevant identity for our problem is:
sin(A) * sin(B) = 1/2 * [cos(A - B) - cos(A + B)]
Let's apply this identity to sin(7π/12) * sin(11π/12). Here, A = 7π/12 and B = 11π/12. So,
sin(7π/12) * sin(11π/12) = 1/2 * [cos(7π/12 - 11π/12) - cos(7π/12 + 11π/12)]
Simplify the angles inside the cosines:
sin(7π/12) * sin(11π/12) = 1/2 * [cos(-4π/12) - cos(18π/12)]
Simplify further:
sin(7π/12) * sin(11π/12) = 1/2 * [cos(-π/3) - cos(3π/2)]
Remember that cos(-θ) = cos(θ), so cos(-π/3) = cos(π/3) = 1/2. Also, cos(3π/2) = 0. Therefore:
sin(7π/12) * sin(11π/12) = 1/2 * [1/2 - 0] = 1/2 * 1/2 = 1/4
Voila! We arrive at the same answer, 1/4, using a different approach. This method is often quicker once you remember the product-to-sum identity. It highlights the power of knowing different trigonometric identities and how they can be used to solve the same problem in various ways. It shows us that there's more than one way to skin a cat (figuratively speaking, of course!). This approach involves fewer steps but requires knowledge of a different set of trigonometric identities. The beauty of mathematics is that often, there are multiple paths to the same solution. The use of product-to-sum identities can greatly simplify calculations. This method provides a clear, concise alternative for solving the problem. The utilization of product-to-sum identities leads to a streamlined solution. Remember, understanding different approaches enhances your problem-solving skills.
Conclusion: Mastering Trigonometric Calculations
So there you have it, guys! We've successfully calculated sin(7π/12) * sin(11π/12) using two different methods. We broke down the problem step-by-step, making sure that every part was clear and easy to understand. We used trigonometric identities like the sum and difference formulas and product-to-sum identities to simplify the expression. The final answer, as we found using both methods, is 1/4. We also showed how different identities can be used to solve the same problem. This demonstrates the flexibility and interconnectedness of trigonometric concepts. The key takeaway here is that trigonometric problems might look complex at first glance, but with the right knowledge and a systematic approach, they become manageable. Regularly practicing trigonometric problems will significantly improve your skills and confidence. You now have the tools to tackle similar problems. We encourage you to try similar problems on your own to solidify your understanding. The ability to manipulate trigonometric expressions is a valuable skill. Understanding the underlying principles makes the learning process much more effective. Both methods gave the same result, confirming the correctness of our calculations. Keep practicing, and you'll become a trigonometry whiz in no time! Remember to always double-check your work and to understand the reasoning behind each step. Now you can solve similar trigonometric problems with confidence. The use of multiple methods also provides a way to verify the correctness of the answer. Congratulations, you've successfully navigated this trigonometric challenge! Keep practicing, and you'll master these concepts in no time. Trigonometry is a fascinating field. Keep up the good work! We've covered the key concepts and techniques necessary to solve this problem effectively. The versatility of trigonometric identities makes them essential tools in mathematics.
