Hey guys! Let's break down this trigonometry problem together. We need to figure out the value of the expression 2 sin(60°) cos(60°). Don't worry, it's not as scary as it looks! We'll go through it step-by-step, and you'll see how easy it is. This is a classic problem that pops up in trigonometry and pre-calculus, so understanding it will definitely help you out. We'll use some well-known trigonometric values and a handy identity to get to the solution. So grab your thinking caps, and let's get started!

Understanding the Problem

First, let's make sure we understand what the problem is asking. We have the expression 2 sin(60°) cos(60°), and we need to find its numerical value. This means we need to know the values of sin(60°) and cos(60°), and then plug them into the expression. Remember, these are standard trigonometric values that you'll often encounter, so it's good to have them memorized or know how to quickly derive them. The angle 60° is a special angle in trigonometry, and its sine and cosine values are well-defined. Knowing these values will allow us to simplify the expression and find the final answer. Essentially, we're going to substitute the sine and cosine values and then perform a simple arithmetic calculation. The key here is to recognize the trigonometric values and apply them correctly. This type of problem is fundamental in understanding how trigonometric functions work, especially when dealing with specific angles. Mastering these types of calculations will build a strong foundation for more complex trigonometric problems in the future. It's all about breaking it down into smaller, manageable steps and understanding the underlying concepts. Are you ready? Let's move on to the next part and find those values!

Finding the Values of sin(60°) and cos(60°)

Alright, let's find those trigonometric values we need. The value of sin(60°) is √3/2, and the value of cos(60°) is 1/2. These are fundamental values that are derived from the unit circle or special right triangles (like the 30-60-90 triangle). If you don't have them memorized, no worries! You can always quickly sketch a 30-60-90 triangle to figure them out. In a 30-60-90 triangle, the side opposite the 30° angle is half the hypotenuse, and the side opposite the 60° angle is √3/2 times the hypotenuse. This relationship allows you to easily determine the sine and cosine values for these angles. So, where do these values come from? Imagine an equilateral triangle with sides of length 2. If you bisect one of the angles, you create two 30-60-90 right triangles. Using the Pythagorean theorem, you can find the lengths of all the sides, and from there, you can calculate the sine and cosine of 60°. This is a handy trick to remember if you ever forget the exact values. The important thing is to understand how these values are derived and to be able to recall them quickly when you need them. Now that we have sin(60°) = √3/2 and cos(60°) = 1/2, we can plug them into our original expression. Let's do that in the next section and see how the calculation unfolds. With these values in hand, the rest of the problem becomes much simpler. Are you ready to substitute and simplify? Let's go!

Substituting the Values

Now that we know sin(60°) = √3/2 and cos(60°) = 1/2, let's substitute these values into our expression: 2 sin(60°) cos(60°). This gives us 2 * (√3/2) * (1/2). See? It's just a matter of plugging in the numbers. The next step is to simplify this expression. We'll multiply the numbers together, being careful to keep track of the square root. When multiplying fractions, you multiply the numerators together and the denominators together. In this case, we have 2 in the numerator and 2 * 2 = 4 in the denominator. The √3 stays in the numerator. So, let's simplify this step-by-step. We start with 2 * (√3/2) * (1/2). First, multiply 2 and √3/2 to get √3. Then, multiply √3 by 1/2 to get √3/2. This is a straightforward calculation, but it's important to be precise to avoid errors. Remember, the order of operations matters, so make sure you're following the correct sequence. In this case, multiplication is associative, so we can multiply in any order we like. The key is to simplify each step as much as possible to make the calculation easier. Now that we've substituted the values and simplified the expression, we're almost there! The next step is to recognize a trigonometric identity that can help us simplify the result even further. Are you ready to see the final trick? Let's move on to the next section!

Recognizing the Trigonometric Identity

Okay, here's a cool trick. The expression 2 sin(θ) cos(θ) is actually a well-known trigonometric identity! It's equal to sin(2θ). Recognizing this identity can make the problem much easier to solve. In our case, θ = 60°, so 2 sin(60°) cos(60°) = sin(2 * 60°) = sin(120°). This simplifies our problem to finding the value of sin(120°). Isn't that neat? Trigonometric identities are equations that are always true for any value of the variable. They're like shortcuts that can save you a lot of time and effort. There are many trigonometric identities, and it's helpful to memorize the most common ones. The double-angle identity for sine is one of the most frequently used identities, and it's definitely worth knowing. So, how do we know that 2 sin(θ) cos(θ) = sin(2θ)? This can be derived from the angle addition formula for sine: sin(A + B) = sin(A) cos(B) + cos(A) sin(B). If we let A = θ and B = θ, then we get sin(θ + θ) = sin(θ) cos(θ) + cos(θ) sin(θ) = 2 sin(θ) cos(θ). Therefore, sin(2θ) = 2 sin(θ) cos(θ). Understanding where these identities come from can help you remember them better. Now that we've recognized the identity and simplified our problem to finding the value of sin(120°), let's do that in the next section. We're almost at the finish line! Are you excited to see the final answer? Let's go!

Finding the Value of sin(120°)

Now we need to find the value of sin(120°). Remember that 120° is in the second quadrant. In the second quadrant, the sine function is positive. We can find the reference angle for 120° by subtracting it from 180°: 180° - 120° = 60°. So, sin(120°) = sin(60°). Since we already know that sin(60°) = √3/2, then sin(120°) = √3/2. And that's it! We've found the value of sin(120°). But let's think about why this works. The unit circle is a helpful tool for understanding trigonometric functions and their values at different angles. The sine of an angle is represented by the y-coordinate of the point on the unit circle corresponding to that angle. Angles that are supplementary (add up to 180°) have the same sine value because they have the same y-coordinate on the unit circle. So, since 120° and 60° are supplementary angles, they have the same sine value. This is a useful fact to remember when working with trigonometric functions. It can help you quickly find the values of angles in different quadrants. Now that we know sin(120°) = √3/2, we can conclude that 2 sin(60°) cos(60°) = √3/2. And that's our final answer! We've successfully solved the problem using trigonometric values and identities. Great job, guys! I hope this explanation was helpful and clear. Remember to practice these types of problems to build your skills and confidence in trigonometry. You've got this!

Therefore, the Final Answer

So, to wrap it all up, the value of 2 sin(60°) cos(60°) is √3/2. We started by finding the values of sin(60°) and cos(60°), then substituted them into the expression. We recognized the trigonometric identity 2 sin(θ) cos(θ) = sin(2θ), which simplified the problem to finding the value of sin(120°). Finally, we used the fact that sin(120°) = sin(60°) to find the answer. This problem demonstrates the importance of knowing your trigonometric values and identities. It also shows how recognizing patterns and using shortcuts can make solving problems much easier. Remember to practice these types of problems regularly to reinforce your understanding. Trigonometry is a fundamental topic in mathematics, and mastering these concepts will be essential for your future studies. I hope this explanation was helpful and clear. If you have any questions, feel free to ask. And remember, keep practicing, and you'll become a trigonometry pro in no time! You've got this!

Final Answer: √3/2