Evaluate Cos(5π/12)cos(π/4) + Sin(5π/12)sin(π/4)
Hey guys! Let's dive into how we can evaluate the trigonometric expression: cos(5π/12)cos(π/4) + sin(5π/12)sin(π/4). This problem might look a bit intimidating at first, but don't worry, we'll break it down step by step. We'll use some trigonometric identities that you might remember from your math classes. The key here is to recognize a familiar pattern that simplifies the whole thing. So, buckle up, and let’s get started! Think about those trig identities – they're our secret weapon for this kind of problem. We're aiming to make this complex expression much easier to handle, and it’s totally achievable with the right approach. Ready to jump in and see how it's done? Let's go!
Recognizing the Cosine Angle Addition Formula
The first thing we should do, guys, is recognize that the expression closely resembles the cosine angle addition formula. This is a crucial step because it transforms what seems like a complicated calculation into a straightforward application of a well-known identity. Remember the formula? It states:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Notice how our given expression, cos(5π/12)cos(π/4) + sin(5π/12)sin(π/4), perfectly matches the right side of the second formula, cos(A - B). This is our 'aha!' moment! By identifying this structure, we can simplify the problem significantly. We can now see that our problem is just a specific case of this general identity. This recognition is super important because it changes the entire approach. Instead of calculating each trigonometric function separately and then combining them, we can use this neat formula to jump straight to the answer. Isn't that cool? So, with this in mind, the next step is to figure out what A and B are in our case. This will allow us to use the formula directly and find the value of the expression. Let's move on to that!
Applying the Formula
Now that we've identified the correct trigonometric identity, let's apply it to our problem. From the expression cos(5π/12)cos(π/4) + sin(5π/12)sin(π/4), we can see that:
- A = 5π/12
- B = π/4
Using the formula cos(A - B) = cos(A)cos(B) + sin(A)sin(B), we can rewrite our expression as:
cos(5π/12 - π/4)
The next step is to subtract the angles. To do this, we need a common denominator. Since 12 is a multiple of 4, we can easily convert π/4 to have a denominator of 12. We multiply both the numerator and the denominator of π/4 by 3, which gives us 3π/12. Now we can rewrite our expression as:
cos(5π/12 - 3π/12)
This simplifies to:
cos(2π/12)
Which further simplifies to:
cos(π/6)
See how much simpler that became? By recognizing the pattern and applying the formula, we've transformed a seemingly complex expression into something much more manageable. The beauty of trigonometry lies in these kinds of simplifications. Now that we've got cos(π/6), the next step is to actually evaluate this cosine value. This is where knowing your special angles comes in handy. We're on the home stretch now! Let's figure out what cos(π/6) actually equals.
Evaluating cos(π/6)
Alright, folks, we've simplified our expression down to cos(π/6). Now, we need to evaluate this. If you remember your unit circle or special triangles, you'll know that π/6 radians is equivalent to 30 degrees. The cosine of 30 degrees, or cos(π/6), is a common value that's worth memorizing. If you've got it memorized, great! If not, no worries – we can quickly recall how to find it.
Think of a 30-60-90 right triangle. The sides are in the ratio 1:√3:2, where the side opposite the 30-degree angle (π/6) is 1, the side adjacent to the 30-degree angle is √3, and the hypotenuse is 2. Cosine is defined as the adjacent side over the hypotenuse. Therefore:
cos(π/6) = Adjacent / Hypotenuse = √3 / 2
So, we've found that cos(π/6) is equal to √3 / 2. This is the final piece of the puzzle! We took a complex-looking expression, used a trigonometric identity to simplify it, and then evaluated the resulting trigonometric function. This process really shows the power of knowing your trig identities and special angles. Now we can confidently say we've solved the problem. It’s always a great feeling when you can break down a challenging problem into smaller, manageable steps, isn’t it? Let's wrap up our solution in the next section.
Final Answer
Okay, everyone, we've reached the end! We started with the expression cos(5π/12)cos(π/4) + sin(5π/12)sin(π/4), recognized the cosine angle subtraction formula, simplified it to cos(π/6), and finally evaluated that to be √3 / 2.
Therefore, the final answer is:
cos(5π/12)cos(π/4) + sin(5π/12)sin(π/4) = √3 / 2
And there you have it! We've successfully evaluated the given trigonometric expression. Remember, the key to tackling these kinds of problems is to recognize patterns, recall your trigonometric identities, and break the problem down into smaller steps. Don’t be intimidated by complex-looking expressions; often, they can be simplified using these techniques. Great job sticking with it, guys! Hopefully, this explanation was clear and helpful. Keep practicing, and you'll become a trig whiz in no time! If you ever run into a similar problem, remember the process we used here: identify the relevant identity, apply it carefully, simplify, and then evaluate. You've got this!