Unlock Cosine Products: 24°, 12°, 48°, 84°

Hey math enthusiasts and puzzle solvers! Today, we’re diving deep into the fascinating world of trigonometry to tackle a particularly neat problem: evaluating the product of cosines: cos 24° cos 12° cos 48° cos 84° . This isn’t just about crunching numbers, guys; it’s about understanding the elegant relationships within trigonometric functions and employing some clever algebraic and trigonometric identities to simplify what looks like a daunting expression. We’ll break down this problem step-by-step, revealing how a little bit of knowledge and strategic manipulation can lead to a surprisingly simple answer. So, grab your calculators (or just your thinking caps!) and let’s get started on this trigonometric adventure. Whether you’re a student preparing for exams, a teacher looking for engaging examples, or just someone who enjoys a good mathematical challenge, you’re in for a treat. We’ll explore some fundamental trigonometric identities that are key to unlocking this puzzle, and by the end, you’ll not only know the answer but also appreciate the beauty and power of these mathematical tools. Get ready to boost your understanding and maybe even impress your friends with your newfound trigonometric prowess!

The Challenge: A Product of Cosines

The problem we’re looking to solve is to find the value of the expression: cos 24° cos 12° cos 48° cos 84° . At first glance, this might seem like something you’d need a powerful calculator for. However, in mathematics, especially in trigonometry, there are often elegant shortcuts and identities that can transform complex-looking problems into manageable ones. Our goal here is to find an exact value, not an approximation. This means we need to use the inherent properties of trigonometric functions and standard angle values. We’ll be rearranging terms, using product-to-sum and sum-to-product identities, and perhaps even leveraging some specific angle values or relationships that might not be immediately obvious. The specific angles – 12°, 24°, 48°, and 84° – are not arbitrary. There’s a pattern here, a doubling relationship in some of the angles, which is a strong hint that we should be looking for ways to exploit that. This kind of problem is a classic example of how seemingly unrelated trigonometric expressions can be simplified using a systematic approach. It tests your familiarity with key identities and your ability to spot patterns. So, let’s get ready to roll up our sleeves and dive into the techniques that will help us conquer this expression. Remember, the journey of solving it is often as rewarding as the final answer itself, as it deepens your understanding of the subject.

Step 1: Rearranging for Symmetry

The first strategic move in tackling the product cos 24° cos 12° cos 48° cos 84° is to rearrange the terms. While the order of multiplication doesn’t change the final product, rearranging can help us group terms that have useful relationships or that fit nicely with trigonometric identities. Let’s reorder them as: cos 12° cos 24° cos 48° cos 84° . Now, pay attention to the angles: 12°, 24°, 48°, 84°. You might notice that 24° is 2 * 12°, 48° is 2 * 24°, and so on. This doubling pattern is a huge clue! However, the 84° term breaks the perfect doubling sequence from 12°. This suggests we might need to find a way to relate 84° to the other angles or use a different identity. Sometimes, adding or subtracting specific angles can reveal hidden relationships. For instance, we know that cos(90° - x) = sin x . Let’s see if we can use this. Notice that 84° = 90° - 6° . This doesn’t immediately seem helpful with our current set of angles. However, what about cos 84° ? Can we relate it to something else? Let’s consider the complementary angle relationship more broadly. We know cos(90° - θ) = sin(θ) . If we look at cos 84° , its complement is 6° . If we look at cos 48° , its complement is 42° . If we look at cos 24° , its complement is 66° . And for cos 12° , its complement is 78° . This direct complementary approach doesn’t seem to simplify things much. Instead, let’s focus on the doubling pattern. We have cos 12° , cos 24° , cos 48° . If we had cos 6° and cos 96° , it might fit a different pattern. The cos 84° term is the outlier. A common strategy when dealing with products of cosines is to use the identity 2 sin x cos x = sin 2x . This identity allows us to convert a product involving cosine into a sine function, often simplifying the expression. To use this, we’d ideally want a sin x term to pair with our cos x terms. Let’s consider multiplying the entire expression by sin 12° . This might seem counter-intuitive, but bear with me. If we multiply by sin 12° , we’ll have to divide by it later to keep the value the same. So, let P = cos 24° cos 12° cos 48° cos 84° . We’ll consider sin 12° * P . This initial rearrangement and thinking about potential identities is crucial for setting up the subsequent steps correctly. It’s all about finding the right angle to approach the problem from!

Step 2: Applying the Double Angle Identity for Sine

Okay, guys, we’ve rearranged our product to cos 12° cos 24° cos 48° cos 84° . Now, let’s bring in our secret weapon: the double angle identity for sine, which states sin(2x) = 2 sin(x) cos(x) . Rearranging this, we get cos(x) = sin(2x) / (2 sin(x)) . This looks a bit messy. A more direct application is to use the identity to introduce sine terms. Let’s multiply our entire expression by 2 sin 12° . We’ll have to divide by 2 sin 12° later. So, let’s look at:

(2 sin 12° cos 12°) cos 24° cos 48° cos 84°

Using the identity sin(2x) = 2 sin(x) cos(x) , the term 2 sin 12° cos 12° becomes sin(2 * 12°) = sin 24° . So now we have:

(sin 24° / 2) cos 24° cos 48° cos 84°

We divided by 2 because we initially multiplied by 2 sin 12° , but we only wanted to use sin 12° as a multiplier. To be precise, let’s multiply the whole expression by sin 12° . To keep the value the same, we must also divide by sin 12° .

So, our expression P = cos 12° cos 24° cos 48° cos 84° becomes:

P = (1 / sin 12°) * (sin 12° cos 12°) * cos 24° cos 48° cos 84°

Now, let’s use 2 sin x cos x = sin 2x . We have sin 12° cos 12° . To use the identity, we need a factor of 2. So, let’s rewrite it as:

P = (1 / sin 12°) * (1/2) * (2 sin 12° cos 12°) * cos 24° cos 48° cos 84°

This simplifies to:

P = (1 / (2 sin 12°)) * sin 24° * cos 24° * cos 48° * cos 84°

Now, look at sin 24° cos 24° . Again, we can apply the double angle identity sin(2x) = 2 sin(x) cos(x) , which means sin(x) cos(x) = sin(2x) / 2 . Applying this to sin 24° cos 24° , we get sin(2 * 24°) / 2 = sin 48° / 2 .

Substituting this back into our expression for P :

P = (1 / (2 sin 12°)) * (sin 48° / 2) * cos 48° * cos 84°

Combining the constants:

P = (1 / 4 sin 12°) * sin 48° * cos 48° * cos 84°

Now, we have sin 48° cos 48° . Once more, we use the identity sin(2x) = 2 sin(x) cos(x) , or sin(x) cos(x) = sin(2x) / 2 . So, sin 48° cos 48° becomes sin(2 * 48°) / 2 = sin 96° / 2 .

Substituting this in:

P = (1 / 4 sin 12°) * (sin 96° / 2) * cos 84°

Combining constants again:

P = (1 / 8 sin 12°) * sin 96° * cos 84°

We’ve successfully eliminated two cosine terms by cleverly introducing sine terms using the double angle identity. This is a common and powerful technique in trigonometry problems involving products of sines and cosines. The key was realizing we needed to introduce the sine term via multiplication and subsequent division. The doubling of angles made this process iterative. Each step simplifies the expression by reducing the number of trigonometric functions, and we’re left with fewer terms to evaluate. The strategy worked like a charm, paving the way for the next step where we’ll deal with the remaining sine and cosine terms.

Step 3: Utilizing Trigonometric Identities for Simplification

Alright, we’ve reached a point where our expression looks like P = (1 / 8 sin 12°) * sin 96° * cos 84° . We still have sin 96° and cos 84° to deal with, along with the sin 12° in the denominator. The next crucial step involves recognizing relationships between these remaining angles and applying further trigonometric identities. Let’s focus on sin 96° and cos 84° . Do these angles have any special relationships? We know that sin(180° - x) = sin x and cos(90° - x) = sin x . Let’s try to simplify sin 96° . Using the identity sin(180° - x) = sin x , we can write sin 96° = sin (180° - 96°) = sin 84° . So, our expression becomes:

P = (1 / 8 sin 12°) * sin 84° * cos 84°

Now, look at sin 84° cos 84° . This is another instance where we can apply the double angle identity sin(2x) = 2 sin(x) cos(x) , or sin(x) cos(x) = sin(2x) / 2 . Applying this to sin 84° cos 84° , we get sin(2 * 84°) / 2 = sin 176° / 2 .

Substituting this back into our expression for P :

P = (1 / 8 sin 12°) * (sin 176° / 2)

Combining constants:

P = (1 / 16 sin 12°) * sin 176°

Now, let’s simplify sin 176° . Using the identity sin(180° - x) = sin x , we can write sin 176° = sin (180° - 176°) = sin 4° . This doesn’t seem to immediately help us cancel out the sin 12° .

Let’s backtrack slightly. Remember we had P = (1 / 8 sin 12°) * sin 96° * cos 84° . Instead of simplifying sin 96° first, let’s consider cos 84° . We know that cos(90° - x) = sin x . So, cos 84° = cos (90° - 6°) = sin 6° . If we substitute this in, we get:

P = (1 / 8 sin 12°) * sin 96° * sin 6°

This still doesn’t look easily solvable. Let’s reconsider the simplification of sin 96° . We found sin 96° = sin 84° . So, the expression was P = (1 / 8 sin 12°) * sin 84° * cos 84° .

What if we use the identity sin(90° + x) = cos x ? sin 96° = sin (90° + 6°) = cos 6° . This also doesn’t seem to directly help with sin 12° .

Let’s revisit the relationship cos 84° . Its complement is 6° , so cos 84° = sin 6° . The expression was P = (1 / 8 sin 12°) * sin 96° * cos 84° . So, P = (1 / 8 sin 12°) * sin 96° * sin 6° .

Now, let’s think about sin 12° and sin 6° . We know that sin 12° = 2 sin 6° cos 6° . If we substitute this into the denominator:

P = (1 / (8 * (2 sin 6° cos 6°))) * sin 96° * sin 6°

P = (1 / (16 sin 6° cos 6°)) * sin 96° * sin 6°

We can cancel out sin 6° :

P = (1 / (16 cos 6°)) * sin 96°

Now, let’s simplify sin 96° . We know sin 96° = sin(180° - 96°) = sin 84° . Also, sin 96° = sin(90° + 6°) = cos 6° . Aha! This is useful.

Substituting sin 96° = cos 6° into our expression:

P = (1 / (16 cos 6°)) * cos 6°

Now, the cos 6° terms cancel out!

P = 1 / 16

Wow! Look at that! By strategically using the double angle identity for sine ( sin 2x = 2 sin x cos x ) multiple times, and then cleverly employing complementary and supplementary angle identities ( cos(90-x) = sin x , sin(180-x) = sin x , sin(90+x) = cos x ), we managed to simplify the entire product down to a simple fraction. The key was to keep applying the sin(2x) identity until we could introduce terms that would cancel out with the original sin 12° (which we expanded using sin 12° = 2 sin 6° cos 6° ). This demonstrates the interconnectedness of trigonometric identities and how they can be used in sequence to solve complex problems.

Step 4: The Final Answer and Verification

We have successfully navigated the intricate path of trigonometric simplification, and the result is 1/16 . This is the exact value of the product cos 24° cos 12° cos 48° cos 84° . It’s quite remarkable how a seemingly complex expression involving specific angles boils down to such a simple, elegant number. The process involved a systematic application of key trigonometric identities: the double angle formula for sine ( sin 2x = 2 sin x cos x ) and properties of complementary and supplementary angles ( cos(90° - x) = sin x , sin(180° - x) = sin x ).

To recap the journey: We started with P = cos 12° cos 24° cos 48° cos 84° . By multiplying and dividing by sin 12° and iteratively applying the double angle identity, we transformed the product step-by-step.

1. P = (1 / (2 sin 12°)) * sin 24° * cos 24° * cos 48° * cos 84°
2. P = (1 / (4 sin 12°)) * sin 48° * cos 48° * cos 84°
3. P = (1 / (8 sin 12°)) * sin 96° * cos 84°

At this stage, we used specific angle properties:

• cos 84° = sin 6° (complementary angle)
• sin 96° = cos 6° (using sin(90+x)=cos x or sin(180-x)=sin x followed by cos(90-x)=sin x leads to the same result)

Substituting these into the expression:

P = (1 / (8 sin 12°)) * cos 6° * sin 6°

Then we used the identity sin 12° = 2 sin 6° cos 6° in the denominator:

P = (1 / (8 * (2 sin 6° cos 6°))) * cos 6° * sin 6°

P = (1 / (16 sin 6° cos 6°)) * cos 6° * sin 6°

Canceling out sin 6° and cos 6° from the numerator and denominator leaves us with:

P = 1/16

Verification: While a full algebraic verification is what we just did, we can also use a calculator to approximate the values and see if they match. cos 12° ≈ 0.9781 cos 24° ≈ 0.9135 cos 48° ≈ 0.6691 cos 84° ≈ 0.1045

Multiplying these values: 0.9781 * 0.9135 * 0.6691 * 0.1045 ≈ 0.0625 . And 1/16 = 0.0625 . The numerical approximation strongly supports our exact result. This consistency gives us high confidence in our answer. It’s always good practice to double-check your work, whether through re-derivation or numerical approximation, especially in complex problems like this. The elegance of the 1/16 result is a testament to the power of trigonometric identities.

Conclusion: The Beauty of Trigonometric Simplification

So there you have it, folks! We’ve successfully tackled the product cos 24° cos 12° cos 48° cos 84° and arrived at the remarkably simple answer of 1/16 . This journey through trigonometric identities showcases the underlying beauty and structure within mathematics. What initially appeared to be a complex calculation requiring a calculator transformed into an elegant proof using fundamental identities. The key takeaways from this problem are the power of:

1. Strategic Rearrangement: Reordering terms can reveal patterns and prepare the expression for specific identities.
2. Double Angle Identity for Sine: sin(2x) = 2 sin(x) cos(x) is incredibly useful for converting products of sine and cosine into single sine functions, especially when angles double.
3. Complementary and Supplementary Angle Identities: Identities like cos(90° - x) = sin x and sin(180° - x) = sin x help relate angles and simplify terms.

Problems like this are fantastic for building intuition and reinforcing your understanding of trigonometric functions. They teach you to look for patterns, to manipulate expressions creatively, and to trust in the power of established mathematical rules. Keep practicing, keep exploring, and you’ll find that many complex mathematical puzzles can be unraveled with the right approach and a bit of persistence. Math is all about connections, and trigonometry provides a beautiful landscape for discovering them. We hope you enjoyed this dive into trigonometric simplification and feel more confident in tackling similar problems. Happy calculating!