Cos 48° Cos 12° Explained

Oct 23, 2025 by Jhon Lennon 26 views

Hey guys! Ever stumbled upon a trigonometry problem that looks a bit intimidating, like calculating the value of cos 48° cos 12°? Don't sweat it! We're going to break this down and make it super clear. This isn't just about crunching numbers; it's about understanding some neat trigonometric identities that can simplify complex expressions. So, grab your thinking caps, and let's dive into the fascinating world of trigonometric products!

The Product-to-Sum Identity: Your New Best Friend

When you see a product of two cosine functions, like cos A cos B, the first thing that should pop into your mind is the product-to-sum identity. This is where the magic happens. This identity allows us to transform a product of trigonometric functions into a sum or difference of other trigonometric functions, which are often easier to work with. The specific identity we'll be using is:

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

See? We've taken a multiplication and turned it into an addition, plus some extra terms. This is a game-changer, especially when dealing with angles that aren't as common as 30°, 45°, or 60°.

Now, let's apply this to our problem: cos 48° cos 12°. Here, we can set A = 48° and B = 12°. Let's plug these values into our identity:

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

This is the crucial step. We've successfully transformed the product into a sum involving two new angles. The beauty of this is that 48° - 12° and 48° + 12° result in angles that are much simpler and more familiar:

48° - 12° = 36° 48° + 12° = 60°

So, our expression now becomes:

cos 48° cos 12° = 1/2 [cos(36°) + cos(60°)]

We're getting closer, folks! We've simplified the problem significantly. Now, all we need are the values for cos(36°) and cos(60°). We know that cos(60°) is a standard value, which is 1/2. The value for cos(36°) might not be as immediately recognizable, but it's a known constant in trigonometry.

Finding the Value of cos(36°)

Okay, so how do we get the value for cos(36°)? This is where it gets a little more involved, but it's totally worth understanding. One common way to derive cos(36°) is by using the properties of a regular pentagon or by employing the golden ratio (phi, φ). Let's sketch out a quick approach using algebraic manipulation and trigonometric identities.

Consider the equation 5θ = 180°. If we let θ = 36°, then 5θ = 180°, which is true. We can rewrite this as 2θ = 180° - 3θ. Now, if we take the cosine of both sides, we get cos(2θ) = cos(180° - 3θ). Using the identity cos(180° - x) = -cos(x), we have cos(2θ) = -cos(3θ).

Now, we can use the double angle and triple angle formulas: cos(2θ) = 2cos²θ - 1 and cos(3θ) = 4cos³θ - 3cosθ.

Substituting these into our equation cos(2θ) = -cos(3θ):

2cos²θ - 1 = -(4cos³θ - 3cosθ) 2cos²θ - 1 = -4cos³θ + 3cosθ

Let x = cosθ. The equation becomes:

2x² - 1 = -4x³ + 3x

Rearranging this into a cubic equation:

4x³ + 2x² - 3x - 1 = 0

We know that θ = 36°, so x = cos(36°). We also know that cos(180°) = -1 is a root of cos(5θ) = cos(180°), which relates to this cubic. We can factor out (x + 1) from the cubic equation, as x = -1 corresponds to θ = 180° (or 5θ = 900°, which is 5 * 180°).

Dividing 4x³ + 2x² - 3x - 1 by (x + 1) gives 4x² - 2x - 1.

So, the equation simplifies to (x + 1)(4x² - 2x - 1) = 0.

This means either x + 1 = 0 (so x = -1, which is cos(180°)) or 4x² - 2x - 1 = 0.

We need to solve the quadratic equation 4x² - 2x - 1 = 0 for x = cos(36°). Using the quadratic formula x = [-b ± sqrt(b² - 4ac)] / 2a:

x = [2 ± sqrt((-2)² - 4 * 4 * -1)] / (2 * 4) x = [2 ± sqrt(4 + 16)] / 8 x = [2 ± sqrt(20)] / 8 x = [2 ± 2sqrt(5)] / 8 x = [1 ± sqrt(5)] / 4

Since 36° is in the first quadrant, its cosine must be positive. Therefore, we take the positive root:

cos(36°) = (1 + sqrt(5)) / 4

There you have it! The value of cos(36°) is (1 + sqrt(5)) / 4. This expression involves the golden ratio, φ = (1 + sqrt(5)) / 2, so cos(36°) = φ / 2.

Putting It All Together: The Final Calculation

Now that we have all the pieces, let's get back to our original expression:

cos 48° cos 12° = 1/2 [cos(36°) + cos(60°)]

Substitute the values we found:

cos 48° cos 12° = 1/2 [(1 + sqrt(5)) / 4 + 1/2]

To add the terms inside the bracket, we need a common denominator, which is 4:

cos 48° cos 12° = 1/2 [(1 + sqrt(5)) / 4 + 2/4] cos 48° cos 12° = 1/2 [(1 + sqrt(5) + 2) / 4] cos 48° cos 12° = 1/2 [(3 + sqrt(5)) / 4]

Finally, multiply by 1/2:

cos 48° cos 12° = (3 + sqrt(5)) / 8

And there you have it! The value of cos 48° cos 12° is (3 + sqrt(5)) / 8. Pretty cool, right? We took a seemingly complex product and simplified it using a fundamental trigonometric identity and a bit of algebraic wizardry to find the value of cos(36°). This shows how powerful these identities are in making complex math problems manageable. Keep practicing, and you'll master these in no time!

Alternative Approaches and Double-Checking

While the product-to-sum identity is the most direct route, it's always good to know there might be other ways or at least ways to double-check your work. For instance, you could use the cosine subtraction and addition formulas, but that would likely be much more convoluted.

Let's consider a sanity check. We know that cos(48°) is slightly less than cos(45°) = sqrt(2)/2 ≈ 0.707. And cos(12°) is very close to cos(0°) = 1, but slightly less. So, cos(48°) * cos(12°) should be a positive value, likely less than 1.

Let's approximate our answer: sqrt(5) is roughly 2.236.

(3 + 2.236) / 8 = 5.236 / 8 ≈ 0.6545.

This value is indeed positive and less than 1, which aligns with our expectations. This quick approximation helps confirm that our calculated value is in the right ballpark.

Another way to think about this is through geometric interpretations, especially involving regular polygons. The angle 36° is fundamental in the construction of a regular pentagon. The value (1 + sqrt(5)) / 4 for cos(36°) is a direct consequence of this geometry. While we focused on the algebraic derivation earlier, understanding the geometric origins can deepen your appreciation for these constants in trigonometry.

Remember, guys, trigonometry is all about patterns and relationships. By learning and applying these identities, you're essentially unlocking shortcuts to solve problems that might otherwise seem incredibly difficult. The product-to-sum and sum-to-product identities are indispensable tools in your mathematical arsenal. They allow you to switch between additive and multiplicative forms, which can be crucial depending on the problem at hand.

So, to recap, we used:

Product-to-Sum Identity: cos A cos B = 1/2 [cos(A - B) + cos(A + B)]
Derivation of cos(36°): Using algebraic manipulation of trigonometric identities, leading to cos(36°) = (1 + sqrt(5)) / 4.
Known Value: cos(60°) = 1/2.

By combining these elements, we arrived at the elegant solution (3 + sqrt(5)) / 8. Don't be discouraged if deriving cos(36°) seems complex at first. It's a standard result often provided in textbooks, but understanding its derivation is incredibly rewarding. Keep exploring, keep practicing, and you'll find that trigonometry becomes less of a challenge and more of an exciting puzzle!

I hope this detailed breakdown makes the calculation of cos 48° cos 12° crystal clear for you all. Happy calculating!