Hey guys! Today, we're diving into a trigonometric problem that might seem a bit daunting at first, but trust me, it's totally manageable. We need to find the value of the expression 2 sin(60) cos(60). This involves understanding some basic trigonometric values and knowing a cool identity that simplifies things a lot. So, let's break it down step by step and make sure we nail it. Understanding trigonometric expressions like 2 sin(60) cos(60) is super useful, especially when you're dealing with problems in physics, engineering, or even computer graphics. These expressions pop up all the time, and knowing how to solve them efficiently can save you a ton of time and effort. Plus, it's a great way to sharpen your math skills!

First off, let's recall some fundamental trigonometric values. We know that sin(60) and cos(60) have specific values that are commonly used. Specifically:

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

These values come from the special right triangles, particularly the 30-60-90 triangle. If you're not super familiar with these, it's a good idea to review them. They are lifesavers in trigonometry! The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse, while the cosine is the ratio of the adjacent side to the hypotenuse. For a 60-degree angle in a 30-60-90 triangle, these ratios give us the values we mentioned above.

Now that we have these values, we can plug them into our expression:

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

Let's simplify this:

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

So, after substituting the values and simplifying, we find that 2 sin(60) cos(60) = √3 / 2. This result is interesting because it also equals sin(60). Isn't that neat?

Using the Double Angle Identity

But wait, there's more! There's a slick trigonometric identity that makes solving this type of problem even easier. It's called the double angle identity for sine, and it states:

sin(2θ) = 2 sin(θ) cos(θ)

Where θ (theta) is an angle. This identity is super handy because it condenses the expression 2 sin(θ) cos(θ) into a single sine function. In our case, θ = 60 degrees.

So, using the double angle identity, we can rewrite our original expression as:

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

Now, we just need to find the value of sin(120). If you remember your unit circle, 120 degrees is in the second quadrant, and it has a reference angle of 60 degrees (since 180 - 120 = 60). In the second quadrant, sine is positive.

Therefore, sin(120) = sin(60) = √3 / 2

Using the double angle identity, we arrive at the same answer: 2 sin(60) cos(60) = √3 / 2. This method is often quicker and less prone to errors once you're comfortable with the identity.

Comparing the Two Methods

Okay, so we solved the problem using two different methods, and both led us to the same answer, which is awesome! Let’s take a moment to compare these approaches.

Direct Substitution

• Pros:
  • Straightforward: This method involves directly substituting the known values of sin(60) and cos(60) into the expression. It's very clear and doesn't require memorizing any special identities.
  • Good for Beginners: If you’re just starting with trigonometry, this method helps reinforce the basic values of sine and cosine for common angles.
• Cons:
  • More Steps: It requires a couple of simplification steps after the substitution, which can be a bit more time-consuming compared to using the double angle identity.
  • Potential for Arithmetic Errors: More steps mean a higher chance of making a small arithmetic mistake.

Double Angle Identity

• Pros:
  • Efficient: This method is quicker once you know the identity sin(2θ) = 2 sin(θ) cos(θ). It transforms the problem into finding the sine of a single angle.
  • Less Calculation: Fewer steps reduce the opportunity for errors.
• Cons:
  • Requires Memorization: You need to remember the double angle identity, which might be challenging for some.
  • Understanding Required: It's not just about memorizing; you also need to understand how to apply the identity correctly.

Why This Matters

You might be wondering, "Okay, I can solve this problem, but why does it even matter?" Great question! Trigonometry, and problems like this one, pop up in all sorts of real-world scenarios. Seriously!

• Physics: When you're analyzing projectile motion, figuring out forces, or dealing with waves, trig functions are your best friends. Knowing how to simplify expressions like 2 sin(θ) cos(θ) can make your calculations much easier.
• Engineering: Engineers use trig all the time to design structures, analyze circuits, and work with signals. Whether it's calculating the angle of a bridge support or designing an antenna, trig is essential.
• Computer Graphics: In computer graphics, trig is used to rotate, scale, and position objects in 3D space. Understanding trig functions helps create realistic and visually appealing graphics.
• Navigation: Pilots and sailors use trig for navigation, determining headings, and calculating distances. It's a fundamental tool for anyone who needs to find their way around.

Practice Problems

Want to get even better at this? Here are a few practice problems you can try:

1. Find the value of 2 sin(30) cos(30).
2. Simplify the expression 2 sin(45) cos(45).
3. What is the value of 2 sin(π/3) cos(π/3)?

Try solving these problems using both methods we discussed: direct substitution and the double angle identity. See which one you prefer and which one helps you get to the answer faster!

Conclusion

Alright, guys, we've covered a lot in this article. We started with the expression 2 sin(60) cos(60), and we found its value using two different methods: direct substitution and the double angle identity. Both methods led us to the same answer, √3 / 2, which is awesome!

We also talked about why understanding trig is important and how it's used in various fields like physics, engineering, computer graphics, and navigation. And finally, we gave you some practice problems to help you master this skill.

So, keep practicing, keep exploring, and remember that trig can be your friend! You've got this!