Unlock The Secret: Cos²48° Cos²12° Simplified

by Jhon Lennon 46 views

Hey math whizzes and curious minds! Today, we're diving deep into the fascinating world of trigonometry to tackle a problem that might look a little intimidating at first glance: cos²48° cos²12°. You might be thinking, "What on earth am I supposed to do with these specific angles?" Well, guys, prepare to be amazed because we're going to break it down step-by-step, making it super clear and, dare I say, even fun! We’ll explore some neat trigonometric identities and transformations that will lead us to a surprisingly simple answer. So, grab your calculators (or just your brains!) and let's get this trigonometric party started!

The Challenge: cos²48° cos²12°

Our mission, should we choose to accept it, is to find the exact value of the expression cos²48° cos²12°. When you first see angles like 48° and 12°, they don't immediately scream "special angles" like 0°, 30°, 45°, 60°, or 90°. This means we can't just pluck values from a standard unit circle. Instead, we need to get clever. The presence of the squares (cos²) also suggests that we might want to employ some power-reducing formulas or perhaps try to manipulate the expression using double-angle or product-to-sum identities. The goal here is to transform these less common angles into something more manageable, ultimately leading us to a numerical result without needing to resort to approximations. It’s like solving a puzzle where each piece is a trigonometric identity, and the final picture is a clean, elegant solution.

Unveiling the Identities: Your Trigonometric Toolkit

To conquer cos²48° cos²12°, we need to equip ourselves with the right tools from our trigonometric arsenal. The most crucial identities we'll be leaning on are:

  1. The Power-Reducing Formula: This is a lifesaver when dealing with squared trigonometric functions. For cosine, it's: cos²(θ) = (1 + cos(2θ)) / 2 This identity is super handy because it converts a squared term into a linear term with a doubled angle, which can often simplify things.

  2. The Product-to-Sum Formula: Sometimes, products of trigonometric functions can be transformed into sums. For cosines, one form is: cos(A)cos(B) = [cos(A - B) + cos(A + B)] / 2 This formula is brilliant for turning a product of cosines into a sum of cosines, which can be easier to evaluate.

  3. Double Angle Identity (for cosine): While not directly used in its product form, knowing cos(2θ) = 2cos²(θ) - 1 or cos(2θ) = cos²(θ) - sin²(θ) is fundamental to understanding how angles relate to their doubles. It often underlies other identities.

  4. Complementary Angle Identity: Remember that cos(θ) = sin(90° - θ). This can be useful for switching between sine and cosine if needed, sometimes simplifying expressions.

By strategically applying these identities, we can systematically break down the complex expression cos²48° cos²12° into simpler, calculable parts. It's all about knowing which tool to use for which job.

Step-by-Step Solution: Cracking the Code

Alright, guys, let's put our identities to work on cos²48° cos²12°. We'll go through this methodically, ensuring every step is clear.

Method 1: Using the Power-Reducing Formula First

This approach involves tackling the squares directly.

  1. Apply the power-reducing formula to both cos²48° and cos²12°:

    • cos²48° = (1 + cos(2 * 48°)) / 2 = (1 + cos(96°)) / 2
    • cos²12° = (1 + cos(2 * 12°)) / 2 = (1 + cos(24°)) / 2

  2. Substitute these back into the original expression: cos²48° cos²12° = [(1 + cos(96°)) / 2] * [(1 + cos(24°)) / 2]

  3. Multiply the fractions: = (1 + cos(96°) + cos(24°) + cos(96°)cos(24°)) / 4

  4. Now, we need to handle the product cos(96°)cos(24°). Let's use the product-to-sum identity: cos(A)cos(B) = [cos(A - B) + cos(A + B)] / 2. Here, A = 96° and B = 24°. cos(96°)cos(24°) = [cos(96° - 24°) + cos(96° + 24°)] / 2 = [cos(72°) + cos(120°)] / 2

  5. We know the values for cos(72°) and cos(120°).

    • cos(72°) is a known value related to the golden ratio, which is (√5 - 1) / 4.
    • cos(120°) = -1/2. Substitute these back: cos(96°)cos(24°) = [((√5 - 1) / 4) + (-1/2)] / 2 = [(√5 - 1 - 2) / 4] / 2 = (√5 - 3) / 8

  6. Substitute this result back into the main expression: = (1 + cos(96°) + cos(24°) + (√5 - 3) / 8) / 4

Wait, this looks like it's getting complicated again with cos(96°) and cos(24°). Let's pause and see if there's a simpler path or if we missed a trick.

Let's Rethink: Maybe a different approach is cleaner for cos²48° cos²12°?

Sometimes, seeing the whole picture differently helps. What if we combined the angles first, or used a complementary angle identity? Let's try using the complementary angle identity to see if we can make the angles relate better.

We know cos(48°) = sin(90° - 48°) = sin(42°). And cos(12°) = sin(90° - 12°) = sin(78°). This doesn't immediately simplify the product cos²48° cos²12°.

What if we rewrite it as (cos 48° cos 12°)²? This looks promising!

Method 2: Squaring the Product

  1. Consider the product cos(48°)cos(12°) first. We can use the product-to-sum formula here: cos(A)cos(B) = [cos(A - B) + cos(A + B)] / 2 Let A = 48° and B = 12°. cos(48°)cos(12°) = [cos(48° - 12°) + cos(48° + 12°)] / 2 = [cos(36°) + cos(60°)] / 2

  2. Now, we know the values for cos(36°) and cos(60°).

    • cos(36°) = (√5 + 1) / 4 (This is another famous value connected to the golden ratio).
    • cos(60°) = 1/2. Substitute these values: cos(48°)cos(12°) = [((√5 + 1) / 4) + (1/2)] / 2 = [(√5 + 1 + 2) / 4] / 2 = (√5 + 3) / 8

  3. Now we need to square this result because the original problem was cos²48° cos²12°, which is equivalent to (cos 48° cos 12°)². cos²48° cos²12° = [(√5 + 3) / 8]²

  4. Expand the square: = ( (√5)² + 2(√5)(3) + 3² ) / 8² = ( 5 + 6√5 + 9 ) / 64 = ( 14 + 6√5 ) / 64

  5. Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 2: = ( 7 + 3√5 ) / 32

Wow! This second method was much cleaner and led us directly to a simplified numerical answer for cos²48° cos²12°. The key was realizing that cos²A cos²B is the same as (cos A cos B)² and then applying the product-to-sum identity.

Double Checking Our Work and Alternative Paths

It's always a good idea to double-check our calculations, especially when dealing with square roots and fractions. Let's quickly review the steps for cos²48° cos²12°.

Method 2 recap:

  • Rewrite as (cos 48° cos 12°)².
  • Use product-to-sum on cos 48° cos 12° to get [cos(36°) + cos(60°)] / 2.
  • Substitute known values: cos(36°) = (√5 + 1) / 4 and cos(60°) = 1/2.
  • This gives [(√5 + 1)/4 + 1/2] / 2 = (√5 + 3) / 8.
  • Square the result: [(√5 + 3) / 8]² = (5 + 6√5 + 9) / 64 = (14 + 6√5) / 64.
  • Simplify to (7 + 3√5) / 32.

The steps seem solid, and the use of known values for cos(36°) and cos(60°) is standard. The result (7 + 3√5) / 32 is the simplified exact value for cos²48° cos²12°.

Could we have used other identities? What about complex numbers?

For those who are more advanced, you might wonder about using Euler's formula (e^(iθ) = cos(θ) + i sin(θ)). We know that cos(θ) = (e^(iθ) + e^(-iθ)) / 2.

So, cos²(θ) = [(e^(iθ) + e^(-iθ)) / 2]² = (e^(2iθ) + 2 + e^(-2iθ)) / 4 = (1 + cos(2θ)) / 2. This just brings us back to the power-reducing formula, so it doesn't offer a shortcut here.

However, the angles 48° and 12° are related to 360°/10 = 36° and 360°/30 = 12°. These angles are related to the construction of a regular decagon and a regular triangle, which indeed involve the golden ratio. The value cos(36°) = (√5 + 1) / 4 is deeply connected to these geometric constructions. This underlying connection explains why the √5 appears in our final answer for cos²48° cos²12°.

The Beauty of Trigonometric Simplification

Solving problems like cos²48° cos²12° isn't just about crunching numbers; it's about appreciating the elegance and interconnectedness of mathematics. We started with an expression involving squares of cosines of seemingly arbitrary angles, and through the strategic application of fundamental trigonometric identities—specifically the product-to-sum formula and the recognition that cos²A cos²B = (cos A cos B)²—we arrived at a neat, exact value: (7 + 3√5) / 32. This simplification highlights how seemingly complex mathematical expressions can often be reduced to simpler forms using the right techniques.

The journey involved transforming a product of squares into a squared product, then using a product-to-sum rule to deal with the product of cosines, and finally evaluating it using known exact values for cos(36°) and cos(60°). The process demonstrates the power of algebraic manipulation combined with trigonometric knowledge. It’s a testament to the beauty of mathematical tools that allow us to untangle intricate problems and reveal underlying structures. So next time you see a trigonometric expression that looks daunting, remember that with the right identities and a bit of step-by-step logic, you can often simplify it beautifully.

Conclusion: You've Mastered cos²48° cos²12°!

So there you have it, folks! We successfully tackled cos²48° cos²12° and found its exact value to be (7 + 3√5) / 32. We saw how using the identity cos²A cos²B = (cos A cos B)² and then applying the product-to-sum formula cos A cos B = [cos(A-B) + cos(A+B)]/2 was the most efficient path. We transformed the problem into calculating [cos(36°) + cos(60°)]²/4, used the known values of cos(36°) = (√5 + 1)/4 and cos(60°) = 1/2, and after some algebraic simplification, arrived at our final answer. It’s a great example of how understanding and applying trigonometric identities can unlock solutions to problems that initially seem complex. Keep practicing, and you'll find yourself navigating the world of trigonometry with confidence and ease!