Calculate: Cos(24°) * Cos(12°) * Cos(48°) * Cos(84°)

by Jhon Lennon 53 views

Hey guys! Today, we're diving into a trigonometric problem that looks a bit intimidating at first glance. We need to find the exact value of the expression: cos(24°) * cos(12°) * cos(48°) * cos(84°). Don't worry; it's not as scary as it seems! We'll break it down step by step, using some handy trigonometric identities to simplify things.

Understanding the Trigonometric Expression

So, when we're faced with a product of cosine functions like this, it's essential to consider using trigonometric identities to simplify the expression. The key here is the double-angle formula and the product-to-sum identities. Let's see how we can apply these to our problem. The double-angle formula is particularly useful because it allows us to relate the cosine of an angle to the cosine of twice that angle, which can help in simplifying products of cosine functions. To successfully navigate and solve this problem, it's important to be super familiar with trigonometric identities, especially those dealing with products and double angles. These identities are like the secret sauce that helps transform complex expressions into something much more manageable. It’s also helpful to recognize patterns and common angles. For example, noticing that 24, 12, 48, and 84 are all multiples of 12 can give us a hint about how to proceed. Breaking down the problem into smaller, more manageable steps makes it less daunting. We're not trying to solve everything at once; instead, we're focusing on simplifying one part of the expression at a time. This approach not only makes the problem easier to handle but also reduces the chance of making mistakes along the way. Remember, patience and a systematic approach are your best friends in trigonometry!

Step-by-Step Solution

Okay, let's get started! Here’s how we can solve it:

  1. Rearrange the terms: cos(12°) * cos(48°) * cos(24°) * cos(84°). This rearrangement will help us pair terms strategically.

  2. Apply the product-to-sum formula: We can use the identity 2*cos(A)cos(B) = cos(A + B) + cos(A - B). First, let's apply it to cos(12°) * cos(48°). So, 2cos(12°) * cos(48°) = cos(12° + 48°) + cos(12° - 48°) = cos(60°) + cos(-36°). Since cos(-x) = cos(x), we have cos(60°) + cos(36°).

    • We know that cos(60°) = 1/2.

    • Also, cos(36°) = (1 + √5) / 4. Therefore, 2*cos(12°) * cos(48°) = 1/2 + (1 + √5) / 4 = (2 + 1 + √5) / 4 = (3 + √5) / 4.

    • So, cos(12°) * cos(48°) = (3 + √5) / 8.

  3. Now, let's deal with cos(24°) * cos(84°): Again, using the product-to-sum formula, 2*cos(24°) * cos(84°) = cos(24° + 84°) + cos(24° - 84°) = cos(108°) + cos(-60°). Since cos(-60°) = cos(60°) = 1/2, we have cos(108°) + 1/2.

    • We also know that cos(108°) = -sin(18°) = -(√5 - 1) / 4. Therefore, 2*cos(24°) * cos(84°) = -(√5 - 1) / 4 + 1/2 = (-√5 + 1 + 2) / 4 = (3 - √5) / 4.

    • So, cos(24°) * cos(84°) = (3 - √5) / 8.

  4. Multiply the results: Now we multiply the two results we obtained: cos(12°) * cos(48°) * cos(24°) * cos(84°) = [(3 + √5) / 8] * [(3 - √5) / 8] = (9 - 5) / 64 = 4 / 64 = 1/16.

Therefore, the value of cos(24°) * cos(12°) * cos(48°) * cos(84°) is 1/16.

Alternative Approach Using Multiple Angle Formulas

Another cool method involves cleverly using multiple angle formulas. This approach can be a bit more direct, especially if you're comfortable manipulating trigonometric identities. Here’s how it goes:

  1. Rewrite the expression: Start with cos(24°) * cos(12°) * cos(48°) * cos(84°).

  2. Multiply and divide by sin(12°): This might seem random, but trust me! We get: [sin(12°) * cos(12°) * cos(24°) * cos(48°) * cos(84°)] / sin(12°).

  3. Apply the double angle formula: Remember that sin(2x) = 2*sin(x)*cos(x). So, sin(12°) * cos(12°) = (1/2) * sin(24°). Our expression now becomes: [(1/2) * sin(24°) * cos(24°) * cos(48°) * cos(84°)] / sin(12°).

  4. Keep applying the double angle formula: (1/2) * sin(24°) * cos(24°) = (1/4) * sin(48°). Now we have: [(1/4) * sin(48°) * cos(48°) * cos(84°)] / sin(12°).

    • Again, (1/4) * sin(48°) * cos(48°) = (1/8) * sin(96°). So, the expression is: [(1/8) * sin(96°) * cos(84°)] / sin(12°).

  5. Use the identity sin(96°) = cos(6°): Because sin(90° + x) = cos(x), we can rewrite sin(96°) as cos(6°). Now our expression is: [(1/8) * cos(6°) * cos(84°)] / sin(12°).

  6. Apply the product-to-sum formula again: 2*cos(6°) * cos(84°) = cos(84° + 6°) + cos(84° - 6°) = cos(90°) + cos(78°) = 0 + cos(78°) = cos(78°).

    • So, cos(6°) * cos(84°) = (1/2) * cos(78°).

    • Our expression becomes: [(1/8) * (1/2) * cos(78°)] / sin(12°) = cos(78°) / (16 * sin(12°)).

  7. Realize that cos(78°) = sin(12°): Since cos(90° - x) = sin(x), cos(78°) = sin(12°). Therefore, the expression simplifies to: sin(12°) / (16 * sin(12°)) = 1/16.

So, using this alternative approach, we also arrive at the same answer: 1/16. Isn't that neat?

Key Trigonometric Identities Used

To solve this problem effectively, we relied on a few key trigonometric identities. Make sure you're familiar with these!

  • Product-to-Sum Formulas: 2*cos(A)*cos(B) = cos(A + B) + cos(A - B)
  • Double-Angle Formula: sin(2x) = 2*sin(x)*cos(x)
  • Complementary Angle Identities: sin(90° - x) = cos(x) and cos(90° - x) = sin(x)
  • Cosine of Supplementary Angles: cos(180° - x) = -cos(x)
  • Even and Odd Properties: cos(-x) = cos(x)

Having these identities in your toolkit makes tackling these kinds of problems much easier. They allow you to transform and simplify expressions, revealing hidden relationships and leading you to the solution. Don't underestimate the power of these identities – they're your best friends in trigonometry!

Common Mistakes to Avoid

When working through trigonometric problems like this, it's easy to make a few common mistakes. Here’s what to watch out for:

  • Incorrectly Applying Identities: Make sure you're using the correct formula and applying it accurately. Double-check the signs and coefficients!
  • Sign Errors: Pay close attention to the signs, especially when dealing with negative angles or using identities involving subtraction.
  • Forgetting Basic Values: Knowing the values of trigonometric functions for common angles (0°, 30°, 45°, 60°, 90°) is crucial. Don't mix them up!
  • Not Simplifying Completely: Always simplify your expression as much as possible. This can reveal hidden cancellations or make further steps easier.
  • Rushing Through the Steps: Take your time and work through each step carefully. Rushing can lead to careless errors.

By being mindful of these potential pitfalls, you can increase your chances of solving the problem correctly and efficiently. Remember, accuracy is key in trigonometry!

Importance of Practice

Like with anything in math, practice makes perfect! The more you work with trigonometric identities and expressions, the more comfortable you'll become. Try solving similar problems to reinforce your understanding. The beauty of trigonometry is that it's all about recognizing patterns and applying the right tools. The more you practice, the better you'll get at spotting these patterns and choosing the appropriate identities to use. Don't be afraid to make mistakes – they're a valuable part of the learning process. Each mistake is an opportunity to understand where you went wrong and how to avoid it in the future. So, grab a pencil, find some practice problems, and start honing your skills! With enough effort, you'll become a trigonometry master in no time!

Conclusion

So, there you have it! The value of cos(24°) * cos(12°) * cos(48°) * cos(84°) is 1/16. We solved it using product-to-sum formulas and a clever trick involving multiple angle formulas. Remember, trigonometry might seem tough at first, but with the right tools and a bit of practice, you can conquer any problem! Keep practicing, and you'll become a trig whiz in no time! Keep up the great work, and happy calculating!