Hey guys! Let's dive into a trigonometric problem that might seem a bit daunting at first, but trust me, it's totally manageable. We're going to figure out the value of the expression 2 sin(60) cos(60). Now, if you're thinking, "Ugh, trigonometry!" don't worry. We'll break it down step-by-step so that everyone, from beginners to seasoned math enthusiasts, can follow along. We will explore the fundamental concepts of trigonometry and provide a detailed, easy-to-understand explanation to solve this problem. By the end of this guide, you’ll not only know the answer but also understand the underlying principles that make it work. So, grab your calculators (or your mental math hats) and let's get started!

Understanding the Basics: Sine and Cosine

Before we jump right into calculating 2 sin(60) cos(60), let's make sure we're all on the same page when it comes to sine and cosine. These are two of the primary trigonometric functions, and they're all about the relationships between the angles and sides of right triangles. Specifically, the sine of an angle (often written as sin(θ)) is the ratio of the length of the side opposite the angle to the length of the hypotenuse. On the flip side, the cosine of an angle (written as cos(θ)) is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. Okay, so what does this mean in practice? Imagine a right triangle with one angle labeled θ. The side opposite θ is the side that doesn't touch the angle. The side adjacent to θ is the side that does touch the angle (but isn't the hypotenuse, which is always the longest side and opposite the right angle). Sine and cosine are fundamental concepts in trigonometry, and understanding them is essential for solving more complex problems. Remember, these functions relate angles to the ratios of sides in a right triangle. This understanding will help you visualize and solve trigonometric equations more effectively.

Special Angles: 60 Degrees

Now that we've refreshed our understanding of sine and cosine, let's focus on the specific angle we're dealing with: 60 degrees. Angles like 30°, 45°, and 60° are often called "special angles" in trigonometry because their sine and cosine values are commonly used and have nice, clean values. For a 60-degree angle, the sine and cosine values are as follows:

• sin(60°) = √3 / 2
• cos(60°) = 1 / 2

These values are derived from the geometry of a 30-60-90 triangle, which is a special type of right triangle. In a 30-60-90 triangle, the side lengths are in a specific ratio, allowing us to easily determine the sine and cosine of the 30 and 60-degree angles. You can either memorize these values or learn how to derive them using the properties of 30-60-90 triangles. Remembering these special angles and their corresponding sine and cosine values will significantly speed up your problem-solving process in trigonometry. These values pop up frequently, so having them at your fingertips is super useful. Understanding where these values come from (the 30-60-90 triangle) also helps in remembering them. So, make sure you're comfortable with these values before moving on.

Solving 2 sin(60) cos(60)

Alright, with the groundwork laid, let's tackle the main problem: finding the value of 2 sin(60) cos(60). We know that:

• sin(60°) = √3 / 2
• cos(60°) = 1 / 2

So, we can substitute these values into our expression:

2 sin(60) cos(60) = 2 * (√3 / 2) * (1 / 2)

Now, let's simplify. First, multiply the numbers together:

2 * (√3 / 2) * (1 / 2) = (2 * √3 * 1) / (2 * 2) = 2√3 / 4

Then, we can simplify the fraction by dividing both the numerator and the denominator by 2:

2√3 / 4 = √3 / 2

And there you have it! The value of 2 sin(60) cos(60) is √3 / 2. This step-by-step approach breaks down the problem into manageable chunks, making it easier to follow and understand. Remember, the key is to substitute the known values of sin(60) and cos(60) and then simplify the resulting expression. This method can be applied to similar trigonometric problems involving special angles. Double-check your calculations to ensure accuracy, and you'll be solving these problems like a pro in no time.

An Alternative Approach: The Double Angle Formula

Now, here's a cool trick! There's a trigonometric identity called the double angle formula for sine, which states that:

sin(2θ) = 2 sin(θ) cos(θ)

Do you notice anything familiar? Our expression, 2 sin(60) cos(60), looks exactly like the right side of this formula! That means we can rewrite our expression as:

2 sin(60) cos(60) = sin(2 * 60) = sin(120)

Now, we need to find the value of sin(120). Since 120 degrees is in the second quadrant, we can use the property that sin(180° - θ) = sin(θ). In this case:

sin(120) = sin(180 - 60) = sin(60)

And we already know that sin(60) = √3 / 2. So, using the double angle formula, we arrive at the same answer: √3 / 2. This approach not only confirms our previous result but also introduces a powerful tool for simplifying trigonometric expressions. The double angle formula can be a real time-saver when you recognize it. It transforms a product of sine and cosine into a single sine function, often making the problem easier to solve. Keep an eye out for opportunities to use these kinds of identities; they can make your life much simpler! Understanding trigonometric identities is a crucial skill in advanced mathematics, and mastering them will help you tackle more complex problems with confidence.

Why is this Important?

You might be wondering, "Okay, I can calculate 2 sin(60) cos(60), but why does it even matter?" Well, trigonometry isn't just some abstract math concept. It has tons of real-world applications. It is used in fields like physics, engineering, computer graphics, and navigation. For example, engineers use trigonometric functions to analyze structures, calculate forces, and design stable buildings and bridges. Physicists use trigonometry to study wave motion, optics, and mechanics. Computer graphics programmers use trigonometry to create realistic 3D models and animations. Navigators use trigonometry to determine positions and directions. The ability to work with trigonometric functions and solve problems like the one we tackled today is a fundamental skill in many technical fields. Understanding these concepts provides a solid foundation for more advanced studies and real-world problem-solving. Moreover, the problem-solving skills you develop by working through trigonometric problems can be applied to other areas of mathematics and science. So, while it might seem like a simple calculation, it's a stepping stone to more complex and fascinating applications.

Conclusion

So, there you have it! The value of 2 sin(60) cos(60) is √3 / 2. We walked through the solution step-by-step, using both direct substitution and the double angle formula. Hopefully, this guide has not only given you the answer but also helped you understand the underlying trigonometric principles. Remember, practice makes perfect! The more you work with sine, cosine, and other trigonometric functions, the more comfortable you'll become. Don't be afraid to tackle challenging problems and explore different approaches. With a little bit of effort, you'll be a trigonometry whiz in no time. And who knows, maybe one day you'll be using these skills to design the next skyscraper or create the next blockbuster movie! Keep exploring, keep learning, and most importantly, keep having fun with math!