Finding Side BC: Cosine Rule In Action

Hey guys! Let's dive into a classic geometry problem. We're going to use the cosine rule to find the length of a side in a triangle. It's super useful for solving triangles when you know certain angles and sides. The problem we're tackling is: We have a triangle ABC, with angle A being 60 degrees. The side AB is 7 cm, and the side AC is 8 cm. Our mission? To calculate the length of side BC. Ready to get started? The cosine rule is a fundamental concept in trigonometry, and understanding it will help you a lot in various math and real-world applications. Let's break it down step by step so it's easy to grasp. This will be so fun and you will love it.

So, the cosine rule essentially relates the lengths of the sides of a triangle to the cosine of one of its angles. The rule states: c² = a² + b² - 2ab cos(C), where a, b, and c are the side lengths, and C is the angle opposite side c. In our specific problem, we're trying to find side BC. So, let's designate side BC as a. Side AB will be c, and side AC will be b. Angle A (60 degrees) is opposite side a (which is BC). We are given two sides and the included angle. Therefore, we can find the side opposite the angle by using the cosine rule.

Now, let's plug in the values! We have AB = 7 cm, AC = 8 cm, and angle A = 60 degrees. We can use the cosine rule which is a² = b² + c² - 2bc cos(A). Let’s substitute those values into the formula. This will allow us to isolate and calculate the unknown side BC. The beauty of the cosine rule lies in its versatility. It's not just a formula; it's a powerful tool that helps us unlock the secrets of triangles when we're given some key pieces of information, like sides and angles. The application of the cosine rule makes it easier to work with these values and understand how they relate to the properties of the triangle. The more we practice using this rule, the more comfortable and confident we will get in our ability to solve different kinds of triangle problems, no sweat.

Applying the Cosine Rule to Solve

Alright, let’s get down to business! We know the formula: a² = b² + c² - 2bc cos(A). Let's substitute the values. Remember, a is the side we want to find (BC), b is AC (8 cm), c is AB (7 cm), and A is 60 degrees. This gives us: a² = 8² + 7² - 2 * 8 * 7 * cos(60°). Let’s do some calculations! We square the sides and multiply the relevant values and then incorporate the value of the cosine angle that we have. A little bit of algebra and you'll get the final result. Cosine of 60 degrees (cos(60°)) is 0.5. Now, plug in the values and begin calculations. This makes the math pretty straightforward! So, our equation becomes a² = 64 + 49 - 2 * 8 * 7 * 0.5. This simplifies to a² = 64 + 49 - 56. Adding and subtracting, we get a² = 57. But hold on, we need to find a (the length of BC), not a². To find a, we take the square root of both sides. This is how we get to the final answer. The square root of 57 is approximately 7.55 cm.

And there you have it! The length of side BC is approximately 7.55 cm. We've successfully used the cosine rule to solve the triangle problem. Isn't that cool? The power of the cosine rule allows us to calculate unknown sides and angles with precision, offering us a window into the properties of triangles. Remember, understanding this is more than just memorizing a formula; it's about seeing how different parts of a triangle are connected and how we can use math to solve them. By applying the cosine rule, we're not just finding a single side; we're unlocking a world of possibilities when it comes to understanding and working with triangles.

Summary of Steps and Key Takeaways

Okay, let's recap what we did, so you're totally clear on how we found the solution. First, we identified the given information: angle A, side AB, and side AC. Then, we realized that we could apply the cosine rule. Remember the cosine rule? a² = b² + c² - 2bc cos(A). It's super important to memorize it! Next, we plugged in our values. a² = 8² + 7² - 2 * 8 * 7 * cos(60°). We calculated the squares, multiplied and figured out that the cosine of 60 degrees is 0.5. Then, we simplified the equation and found a² = 57. Finally, we took the square root to find a (which is the length of BC), and we got around 7.55 cm. Amazing! The cosine rule is a versatile tool and is super important. We can use it when we know two sides and the included angle. When you start working on these problems, you'll see how the cosine rule is invaluable for various situations, such as architecture, engineering, and navigation. So keep practicing and you'll find it gets easier every time.

The ability to use the cosine rule opens up a wide range of problem-solving opportunities. Imagine calculating distances, angles, and areas, making it a valuable tool in fields like surveying and physics. The rule provides an elegant way to deal with triangles, which are fundamental shapes found everywhere in the world around us. With each problem, your understanding of trigonometry and geometry will grow, and you'll be able to visualize and solve complex problems with ease. This journey will provide you with a deeper appreciation for the beauty and power of mathematical concepts. Awesome!