Trigonometry: Cosine Product Identities Explained

Hey math enthusiasts! Ever found yourself staring at a complex trigonometric expression and wishing there was a simpler way to tackle it? Well, buckle up, because today we’re diving deep into the world of cosine product identities , specifically focusing on the intriguing problem of evaluating cos 24° cos 12° cos 48° cos 84° . This isn’t just about crunching numbers, guys; it’s about unlocking the elegant secrets hidden within trigonometric functions and appreciating the beauty of mathematical simplification. We’ll break down this problem step-by-step, revealing how these seemingly random angles connect through powerful identities. Prepare to be amazed as we transform a challenging product into a much more manageable form. Whether you’re a student gearing up for exams or just a curious mind exploring the wonders of mathematics, this guide is for you. We’ll make sure to explain every concept clearly, so no one gets left behind. So, grab your favorite beverage, get comfy, and let’s embark on this mathematical adventure together! Get ready to boost your trigonometry game and impress your friends with your newfound knowledge.

Unlocking the Power of Product-to-Sum Formulas

Alright, so the core of solving cos 24° cos 12° cos 48° cos 84° lies in understanding and applying the product-to-sum formulas . These are absolute game-changers in trigonometry. Essentially, they allow us to convert a product of trigonometric functions into a sum or difference of other trigonometric functions. This is super useful because sums and differences are often much easier to handle, especially when dealing with multiple terms. The specific identity we’ll be leaning on is:

2 cos A cos B = cos(A + B) + cos(A - B)

Or, rearranged:

cos A cos B = 1/2 [cos(A + B) + cos(A - B)]

See? We take a product of two cosines and turn it into a sum of two cosines. This can be applied repeatedly to simplify our original expression. The key is to strategically pair up the cosine terms in a way that makes the resulting angles convenient to work with. For our problem, cos 24° cos 12° cos 48° cos 84° , we have four terms. We can start by pairing any two. Let’s try pairing cos 24° and cos 12° first. Applying the formula, we get:

cos 24° cos 12° = 1/2 [cos(24° + 12°) + cos(24° - 12°)]

= 1/2 [cos 36° + cos 12°]

Now, our original expression looks like this:

1/2 [cos 36° + cos 12°] * cos 48° cos 84°

We could continue applying the product-to-sum formula to cos 48° cos 84° as well, or we could distribute the 1/2 [cos 36° + cos 12°] term. However, there’s a more elegant path that involves strategic rearrangement and the use of complementary angle identities, which we’ll explore next. The beauty of these formulas is their versatility; you can often find multiple ways to apply them, and the most efficient method usually involves a bit of foresight.

Strategic Rearrangement and Complementary Angles

Now, let’s talk about strategy. While we could just keep blindly applying the product-to-sum formula, math often rewards a bit of clever thinking. Notice the angles in our original expression: 12°, 24°, 48°, and 84°. There’s a pattern here, but it’s not immediately obvious. Often, in these kinds of problems, the solution becomes much clearer if we can arrange the terms so that we end up with angles that are related by complementary relationships (angles that add up to 90°). Recall that cos θ = sin (90° - θ) . This identity is your best friend when dealing with seemingly awkward angles. Let’s look at our angles again: 12°, 24°, 48°, 84°.

Consider the term cos 84° . We know that cos 84° = sin (90° - 84°) = sin 6° . This might not seem immediately helpful, but let’s keep it in mind. What about cos 48° ? It’s sin (90° - 48°) = sin 42° . And cos 12° is sin (90° - 12°) = sin 78° . This transformation isn’t simplifying things dramatically on its own yet. The real magic happens when we pair terms cleverly before applying the product-to-sum identity. Let’s rewrite our expression:

cos 84° cos 48° cos 24° cos 12°

Instead of pairing cos 24° and cos 12° first, let’s try pairing cos 84° and cos 12° . Why? Because 84° + 12° = 96° and 84° - 12° = 72° . Not super helpful. How about cos 84° and cos 24° ? Sum is 108°, difference is 60°. Ah, cos 60° is a known value ( ¹ ⁄ ₂ )! This looks promising. Let’s try this pairing:

cos 84° cos 24° = 1/2 [cos(84° + 24°) + cos(84° - 24°)]

= 1/2 [cos 108° + cos 60°]

= 1/2 [cos 108° + 1/2]

Now our expression is:

[1/2 (cos 108° + 1/2)] * cos 48° cos 12°

This still involves cos 108° , which isn’t a standard angle. Let’s rethink the pairing. What if we pair cos 84° with cos 48° ?

cos 84° cos 48° = 1/2 [cos(84° + 48°) + cos(84° - 48°)]

= 1/2 [cos 132° + cos 36°]

Still not hitting nice known values immediately. The trick here is often to introduce a factor that helps cancel out or simplifies later. Let’s consider multiplying and dividing by sin 12° .

Original expression: P = cos 12° cos 24° cos 48° cos 84°

Multiply by sin 12° :

sin 12° * P = sin 12° cos 12° cos 24° cos 48° cos 84°

We know the double angle identity: sin 2θ = 2 sin θ cos θ . So, sin 12° cos 12° = 1/2 sin (2 * 12°) = 1/2 sin 24° .

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

Again, apply the double angle identity to sin 24° cos 24° : sin 24° cos 24° = 1/2 sin (2 * 24°) = 1/2 sin 48° .

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

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

Apply the double angle identity to sin 48° cos 48° : sin 48° cos 48° = 1/2 sin (2 * 48°) = 1/2 sin 96° .

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

sin 12° * P = 1/8 sin 96° cos 84°

Now, we use the complementary angle identity. We know sin 96° = sin (180° - 96°) = sin 84° . Alternatively, sin 96° = sin (90° + 6°) = cos 6° . Neither seems directly helpful yet. Let’s use the fact that sin 96° = sin (180° - 96°) = sin 84° .

So, sin 12° * P = 1/8 sin 84° cos 84° .

Apply the double angle identity one last time to sin 84° cos 84° : sin 84° cos 84° = 1/2 sin (2 * 84°) = 1/2 sin 170° .

sin 12° * P = 1/8 * (1/2 sin 170°)

sin 12° * P = 1/16 sin 170° .

We know that sin 170° = sin (180° - 170°) = sin 10° . So,

sin 12° * P = 1/16 sin 10° .

This approach is leading us to angles that don’t immediately resolve. Let’s backtrack and try a different pairing using the product-to-sum identity, perhaps focusing on angles that add up to 90 degrees or multiples thereof. The strategy of multiplying by sin 12° seemed promising because it generated double angles, but the final result wasn’t immediately obvious. This highlights that sometimes, the first approach isn’t the most direct, and persistence or trying a different angle is key.

The Elegant Solution: Utilizing cos(60° - x) and cos(60° + x)

Okay, guys, let’s try a different angle (pun intended!). We’re looking at cos 12° cos 24° cos 48° cos 84° . A really neat trick in trigonometry involves recognizing patterns related to cos(60° - x) , cos x , and cos(60° + x) . The identity is:

cos x * cos(60° - x) * cos(60° + x) = 1/4 cos(3x) .

This identity is incredibly powerful for simplifying products of three cosine terms with specific angular relationships. Let’s see if we can make our expression fit this pattern. We have angles 12°, 24°, 48°, 84°.

Let’s try setting x = 12° . Then:

• x = 12°
• 60° - x = 60° - 12° = 48°
• 60° + x = 60° + 12° = 72°

Our expression has cos 12° and cos 48° . It also has cos 24° and cos 84° . This doesn’t directly match the cos x , cos(60-x) , cos(60+x) structure perfectly with just the first three terms. However, let’s rearrange our terms:

P = (cos 12° cos 48°) * (cos 24° cos 84°)

Let’s evaluate the first pair cos 12° cos 48° using the product-to-sum:

cos 12° cos 48° = 1/2 [cos(48° + 12°) + cos(48° - 12°)]

= 1/2 [cos 60° + cos 36°]

= 1/2 [1/2 + cos 36°]

Now let’s look at the second pair cos 24° cos 84° :

cos 24° cos 84° = 1/2 [cos(84° + 24°) + cos(84° - 24°)]

= 1/2 [cos 108° + cos 60°]

= 1/2 [cos 108° + 1/2]

So, our product P becomes:

P = [1/2 (1/2 + cos 36°)] * [1/2 (cos 108° + 1/2)]

P = 1/4 * (1/2 + cos 36°) * (cos 108° + 1/2)

We know that cos 108° = cos (180° - 72°) = -cos 72° . Also, cos 72° = sin (90° - 72°) = sin 18° . And cos 36° is related to the golden ratio, but let’s see if we can simplify using complementary angles more directly.

Let’s use cos 108° = -cos 72° . And we also know cos 72° = sin 18° . And cos 36° = 1 - 2 sin² 18° . This is getting complicated with specific values. Let’s try rearranging the original terms differently to fit the cos x * cos(60-x) * cos(60+x) pattern.

Consider the angles: 12°, 24°, 48°, 84°. Let’s try pairing cos 24° with cos(60°-24°) and cos(60°+24°) . Wait, that’s not right. The angles in the identity are x , 60-x , 60+x . If x=12° , the angles are 12°, 48°, 72°. We have 12° and 48°. We don’t have 72°.

Let’s try x=24° . Then angles are 24°, 60-24=36° , 60+24=84° . We have 24° and 84°. We don’t have 36°.

This implies the cos x cos(60-x) cos(60+x) identity might not be the direct path here without manipulation.

Let’s revisit the multiplication by sin 12° strategy, but pay closer attention to the final steps.

We reached: sin 12° * P = 1/8 sin 96° cos 84°

Now, use cos 84° = sin(90° - 84°) = sin 6° . And sin 96° = sin(90° + 6°) = cos 6° .

So, sin 12° * P = 1/8 * cos 6° * sin 6° .

Using the double angle formula sin 2θ = 2 sin θ cos θ , we have sin 6° cos 6° = 1/2 sin (2 * 6°) = 1/2 sin 12° .

Substituting this back:

sin 12° * P = 1/8 * (1/2 sin 12°)

sin 12° * P = 1/16 sin 12° .

Since sin 12° is not zero, we can divide both sides by sin 12° :

P = 1/16 .

Wow! See how using complementary angles and the double angle formula in a specific sequence led us to a clean numerical answer? This method, while it might seem tricky at first, is incredibly efficient. The key was recognizing that we could transform the product into terms involving sin 6° , cos 6° , and eventually sin 12° , which allowed for cancellation.

The Final Answer and Takeaways

So, after navigating through the twists and turns of trigonometric identities, we’ve arrived at the solution for cos 24° cos 12° cos 48° cos 84° . The value is ¹ ⁄ ₁₆ . Isn’t that neat? A seemingly complex product of four cosine terms simplifies down to a simple fraction. This problem beautifully demonstrates the power and elegance of trigonometric identities.

Key takeaways from this exploration:

1. Product-to-Sum Formulas: Remember cos A cos B = 1/2 [cos(A + B) + cos(A - B)] . These are fundamental for simplifying products.
2. Double Angle Formulas: The sin 2θ = 2 sin θ cos θ identity is your best friend for creating opportunities for cancellation, especially when you can manipulate the expression to involve sin θ cos θ .
3. Complementary Angle Identities: cos θ = sin(90° - θ) and sin θ = cos(90° - θ) are crucial for relating angles and finding hidden simplifications. Also, identities like sin(90° + θ) = cos θ and cos(90° + θ) = -sin θ can be very handy.
4. Strategic Multiplication/Division: Sometimes, multiplying and dividing by a sine term (like sin 12° in our case) can cleverly set up the double angle formulas needed for simplification.
5. Persistence and Pattern Recognition: Don’t be discouraged if the first path you try doesn’t immediately yield a simple answer. Look for patterns, try rearranging terms, and experiment with different identities. The cos x cos(60-x) cos(60+x) = 1/4 cos(3x) identity is also a powerful tool to keep in your arsenal, though it required careful alignment in this specific problem.

This journey into evaluating cos 24° cos 12° cos 48° cos 84° wasn’t just about finding a number; it was about understanding the underlying structure of trigonometry and appreciating how different rules work together harmoniously. Keep practicing, keep exploring, and you’ll find that these seemingly daunting problems become increasingly manageable and even enjoyable. Happy calculating, everyone!