Unlock The Value Of Cos 24° Cos 12° Cos 48° Cos 84°
Hey math whizzes and curious minds! Today, we're diving deep into the fascinating world of trigonometry to unravel the mystery behind the product: cos 24° cos 12° cos 48° cos 84°. You might look at this string of cosine values and think, "What on earth is this going to equal?" Well, buckle up, because we're going to break it down step-by-step, using some super cool trigonometric identities that will make this whole thing click. We'll explore the value of this expression and see how seemingly complex problems can be simplified with the right tools. So, let's get started and see if we can find that neat, tidy answer!
The Magic of Trigonometric Identities
Alright guys, when you see a product of trigonometric functions like our friend, cos 24° cos 12° cos 48° cos 84°, your spidey senses should start tingling for identities. Specifically, the product-to-sum and double-angle formulas are often our best pals in these situations. The double-angle formula for cosine is something like cos(2x) = 2cos²(x) - 1 or cos(2x) = cos²(x) - sin²(x) or cos(2x) = 1 - 2sin²(x). While these are useful, the identity that will really shine here is the product-to-sum identity, or even better, a clever manipulation involving the sine function. You see, there's a beautiful relationship between sine and cosine: sin(2x) = 2sin(x)cos(x). Rearranging this, we get cos(x) = sin(2x) / (2sin(x)). This little trick is pure gold! It allows us to convert a cosine term into a ratio involving sines, which can then often lead to cancellations. Our goal is to find the value of cos 24° cos 12° cos 48° cos 84°, and this identity is our secret weapon. We'll be strategically multiplying and dividing by sine terms to create these sin(2x) forms, paving the way for simplification. Keep your eyes peeled, because the cancellations are where the real magic happens, transforming a complicated product into a simple numerical result. The journey to the value involves seeing these patterns and applying the identities with precision.
Step-by-Step Simplification
Let's get our hands dirty and start simplifying cos 24° cos 12° cos 48° cos 84°. We'll begin by focusing on a few terms and see how we can apply our cos(x) = sin(2x) / (2sin(x)) trick. It's often useful to start with the smaller angles or angles that seem related by a factor of two. Notice that 48° is 2 * 24°, and 24° is 2 * 12°. This relationship is a huge hint that the double-angle identity, or its sine counterpart, will be key. Let's rearrange our expression to make it easier to work with: (cos 12° cos 24° cos 48° cos 84°). Now, let's introduce a sine term. Multiply and divide by sin(12°):
(1 / sin 12°) * (sin 12° cos 12°) * cos 24° cos 48° cos 84°
Using the identity sin(2x) = 2sin(x)cos(x), we can rewrite sin 12° cos 12° as (1/2) sin(2 * 12°) = (1/2) sin 24°. So now we have:
(1 / sin 12°) * (1/2) sin 24° * cos 24° cos 48° cos 84°
Let's pull the 1/2 out:
(1/2) * (1 / sin 12°) * (sin 24° cos 24°) * cos 48° cos 84°
Again, we can use sin(2x) = 2sin(x)cos(x) on sin 24° cos 24°. This becomes (1/2) sin(2 * 24°) = (1/2) sin 48°. Substituting this back in:
(1/2) * (1 / sin 12°) * (1/2) sin 48° * cos 48° cos 84°
Combine the constants:
(1/4) * (1 / sin 12°) * (sin 48° cos 48°) * cos 84°
And again, sin 48° cos 48° becomes (1/2) sin(2 * 48°) = (1/2) sin 96°:
(1/4) * (1 / sin 12°) * (1/2) sin 96° * cos 84°
Pull out the new constant 1/2:
(1/8) * (1 / sin 12°) * sin 96° * cos 84°
Now, we need to deal with sin 96° * cos 84°. This looks a bit tricky, but remember that sin(180° - x) = sin(x) and cos(90° - x) = sin(x). Let's use the complementary angle identity cos(84°) = sin(90° - 84°) = sin(6°). This doesn't immediately help with sin 96°. However, we also know that sin(180° - x) = sin(x). So, sin(96°) = sin(180° - 96°) = sin(84°).
Let's substitute this:
(1/8) * (1 / sin 12°) * sin 84° * cos 84°
Now, sin 84° cos 84° is (1/2) sin(2 * 84°) = (1/2) sin 170°. So we have:
(1/8) * (1 / sin 12°) * (1/2) sin 170°
Which simplifies to:
(1/16) * (1 / sin 12°) * sin 170°
We know that sin(170°) = sin(180° - 170°) = sin(10°). So we have:
(1/16) * (1 / sin 12°) * sin 10°
Now it looks like we have sin 10° / sin 12°. This doesn't immediately simplify to a nice number. Hmm, did we miss something or take a less optimal path? Let's rethink the order or perhaps use a different identity.
A More Elegant Approach
Let's try a slightly different path, focusing on the angles and their relationships. We have cos 12° cos 24° cos 48° cos 84°. Notice that 84° = 90° - 6°, so cos 84° = sin 6°. Also, 48° = 60° - 12° and 24° = 60° - 36°. This might be getting complicated. Let's stick to the double angle and sine manipulation.
Consider the expression again: P = cos 12° cos 24° cos 48° cos 84°.
Let's multiply by sin 12° and divide by sin 12°:
P = (1 / sin 12°) * sin 12° cos 12° cos 24° cos 48° cos 84°
P = (1 / sin 12°) * (1/2) sin 24° cos 24° cos 48° cos 84°
P = (1 / (2 sin 12°)) * (1/2) sin 48° cos 48° cos 84°
P = (1 / (4 sin 12°)) * (1/2) sin 96° cos 84°
P = (1 / (8 sin 12°)) * sin 96° cos 84°
Now, here's a crucial insight. We know that sin 96° = sin (180° - 96°) = sin 84°. Let's substitute that in:
P = (1 / (8 sin 12°)) * sin 84° cos 84°
Using the double angle identity sin(2x) = 2sin(x)cos(x) again, sin 84° cos 84° = (1/2) sin (2 * 84°) = (1/2) sin 170°.
P = (1 / (8 sin 12°)) * (1/2) sin 170°
P = (1 / (16 sin 12°)) * sin 170°
And we know sin 170° = sin (180° - 170°) = sin 10°.
P = (1 / (16 sin 12°)) * sin 10°
P = sin 10° / (16 sin 12°)
This still doesn't look like a simple number. Let's pause and re-evaluate. What if we used the relation cos(x) = sin(90 - x) earlier?
Original expression: cos 12° cos 24° cos 48° cos 84°
Let's rewrite cos 84° as sin (90° - 84°) = sin 6°.
So we have cos 12° cos 24° cos 48° sin 6°. This also doesn't immediately simplify.
Let's consider the angles: 12°, 24°, 48°, 84°. Notice a pattern here: 12, 212, 412, 7*12? Not quite. How about 12, 24, 48, and then 84? The relationship isn't a simple doubling all the way.
There's a common trick for products of cosines like this. If we have cos(theta) cos(2*theta) cos(4*theta) ... cos(2^(n-1)*theta), we can multiply by sin(theta). Let's see if our angles fit this.
We have cos(12°). Then cos(2*12°) = cos(24°). Then cos(2*24°) = cos(48°). BUT the next angle is cos(84°), not cos(96°). This is where the problem deviates from the simplest form.
Let's try pairing angles differently. Consider the identity cos(A)cos(B) = 1/2 [cos(A-B) + cos(A+B)]. This can get messy.
What if we use the identity cos(60° - x)cos(x)cos(60° + x) = 1/4 cos(3x)?
Let's see if our angles fit this. If x = 12°, then cos(12°). cos(60° - 12°) = cos(48°). cos(60° + 12°) = cos(72°). We don't have cos(72°).
Let's rearrange the original expression: cos 12° cos 24° cos 48° cos 84°.
Consider cos 84°. cos 84° = cos(60° + 24°).
Consider cos 48° = cos(60° - 12°).
Consider cos 24°.
Consider cos 12°.
This grouping doesn't seem to directly fit the cos(60-x)cos(x)cos(60+x) identity.
Let's go back to the sine manipulation, it's usually the most robust method.
P = cos 12° cos 24° cos 48° cos 84°
Multiply by sin 12°:
P * sin 12° = sin 12° cos 12° cos 24° cos 48° cos 84°
= (1/2) sin 24° cos 24° cos 48° cos 84°
= (1/4) sin 48° cos 48° cos 84°
= (1/8) sin 96° cos 84°
Now, we need to deal with sin 96° cos 84°.
Using sin(96°) = sin(180° - 96°) = sin(84°).
So, P * sin 12° = (1/8) sin 84° cos 84°
= (1/16) sin(2 * 84°) = (1/16) sin(170°)
= (1/16) sin(180° - 170°) = (1/16) sin(10°)
So, P * sin 12° = (1/16) sin 10°.
This gives P = sin 10° / (16 sin 12°). This is where many people get stuck if they don't see the final step.
Let's re-examine the angles. We have 10° and 12°. Is there a relation we missed?
Wait, let's check the original angles again. 12°, 24°, 48°, 84°.
What if we used the identity cos(x)cos(60-x)cos(60+x) = 1/4 cos(3x) in a different way?
Let's try to express our angles in terms of 60 degrees.
cos 84° = cos(60° + 24°)
cos 48° = cos(60° - 12°)
So we have cos 12° * cos 24° * cos(60° - 12°) * cos(60° + 24°). This is not the form we want.
Let's try another rearrangement of the expression: cos 12° cos 24° cos 48° cos 84°.
Consider the product cos 12° cos 48°.
And cos 24° cos 84°.
Let's use the identity cos A cos B = 1/2 [cos(A-B) + cos(A+B)].
cos 12° cos 48° = 1/2 [cos(48-12) + cos(48+12)] = 1/2 [cos 36° + cos 60°].
cos 24° cos 84° = 1/2 [cos(84-24) + cos(84+24)] = 1/2 [cos 60° + cos 108°].
So the product becomes:
[1/2 (cos 36° + cos 60°)] * [1/2 (cos 60° + cos 108°)]
= 1/4 (cos 36° + 1/2) * (1/2 + cos 108°)
We know cos 108° = cos(180° - 72°) = -cos 72°.
And cos 72° = sin 18°. The value of sin 18° is (sqrt(5) - 1) / 4.
So cos 108° = - (sqrt(5) - 1) / 4 = (1 - sqrt(5)) / 4.
And cos 36° = (sqrt(5) + 1) / 4.
Substitute these values:
1/4 [ (sqrt(5) + 1) / 4 + 1/2 ] * [ 1/2 + (1 - sqrt(5)) / 4 ]
= 1/4 [ (sqrt(5) + 1 + 2) / 4 ] * [ (2 + 1 - sqrt(5)) / 4 ]
= 1/4 [ (sqrt(5) + 3) / 4 ] * [ (3 - sqrt(5)) / 4 ]
= 1/4 * [ (3 + sqrt(5)) * (3 - sqrt(5)) ] / 16
This is a difference of squares: (a+b)(a-b) = a² - b².
= 1/4 * [ 3² - (sqrt(5))² ] / 16
= 1/4 * [ 9 - 5 ] / 16
= 1/4 * 4 / 16
= 1/4 * 1/4
= 1/16
Boom! We got the value! This method, using product-to-sum identities and known values of specific angles, worked beautifully. It shows that sometimes you need to try different identities or rearrange terms to find the path of least resistance. The key was recognizing that cos 36°, cos 60°, and cos 72° (related to cos 108°) have known exact values involving the golden ratio. This is why understanding those special angle values is so important in trigonometry!
The Final Answer and Its Significance
So, after navigating through the twists and turns of trigonometric identities, we've arrived at the magnificent value of cos 24° cos 12° cos 48° cos 84°, which is 1/16. Isn't that neat? It's amazing how a seemingly complex product of four cosine terms boils down to such a simple fraction. This problem is a fantastic illustration of the power and elegance of trigonometry. It highlights how different trigonometric identities can be used in conjunction to solve problems. We saw the application of the product-to-sum formula, the use of complementary angles (cos(90-x) = sin(x)), supplementary angles (cos(180-x) = -cos(x)), and the crucial knowledge of exact values for angles like 36° and 72° (which are related to the golden ratio). These are the moments in math that make you appreciate the underlying structure and beauty. The value 1/16 isn't just a random number; it's the result of applying fundamental mathematical principles accurately. Whether you're preparing for exams, delving into physics, or just enjoying the intellectual puzzle, understanding these simplification techniques is invaluable. It builds confidence and a deeper appreciation for the interconnectedness of mathematical concepts. Keep practicing, keep exploring, and you'll find that many complex trigonometric expressions can be tamed into simple, elegant answers. The journey to find the value of cos 24° cos 12° cos 48° cos 84° is complete, and the answer is a sweet 1/16!
