Calculating 12cos(48°)cos(12°)sin(18°) Simply
Hey guys! Today, we're diving into a fun trigonometric problem: calculating the exact value of the expression 12cos(48°)cos(12°)sin(18°). This might look intimidating at first glance, but don't worry! We'll break it down step by step, using trigonometric identities to simplify it. So, grab your calculators (though we'll try to avoid them as much as possible!) and let's get started!
Understanding the Basics
Before we jump into the problem, let's brush up on some basic trigonometric concepts. First, remember what cosine (cos) and sine (sin) represent on the unit circle. Cosine corresponds to the x-coordinate, and sine corresponds to the y-coordinate. Knowing these fundamental definitions will help you visualize the values we are working with. Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. These identities are essential tools for simplifying complex expressions. We'll be using the product-to-sum formulas, double-angle formulas, and possibly some others along the way. The goal here is to manipulate the given expression into a simpler form that we can easily evaluate. This often involves combining terms, canceling out factors, and using identities to rewrite trigonometric functions in more convenient ways. Practice makes perfect! The more you work with trigonometric functions and identities, the more comfortable you'll become with them. Don't be afraid to experiment and try different approaches. With a bit of patience and persistence, you'll be able to solve even the most challenging problems. Remember, trigonometry is a powerful tool that can be used to model and solve a wide range of problems in physics, engineering, and other fields. By mastering the fundamentals, you'll be well-equipped to tackle these challenges. So, let's get started and see how we can simplify this expression!
Strategy
Our main strategy here involves using trigonometric identities to simplify the expression. A particularly helpful identity for products of trigonometric functions is the product-to-sum formula. We'll also be trying to manipulate the angles to make them more manageable. In our expression 12cos(48°)cos(12°)sin(18°), notice the product of two cosine functions: cos(48°)cos(12°). We can use the product-to-sum identity to rewrite this product as a sum of cosine functions. The product-to-sum identity we will use is:
cos(A)cos(B) = 1/2[cos(A + B) + cos(A - B)]
Applying this identity to cos(48°)cos(12°), we get:
cos(48°)cos(12°) = 1/2[cos(48° + 12°) + cos(48° - 12°)] cos(48°)cos(12°) = 1/2[cos(60°) + cos(36°)]
Now, we know that cos(60°) = 1/2. So, we have:
cos(48°)cos(12°) = 1/2[1/2 + cos(36°)]
Substitute this back into our original expression:
12cos(48°)cos(12°)sin(18°) = 12 * 1/2[1/2 + cos(36°)] * sin(18°) 12cos(48°)cos(12°)sin(18°) = 6[1/2 + cos(36°)]sin(18°) 12cos(48°)cos(12°)sin(18°) = 3sin(18°) + 6cos(36°)sin(18°)
Now, we need to simplify 6cos(36°)sin(18°). We can use another product-to-sum identity:
sin(A)cos(B) = 1/2[sin(A + B) + sin(A - B)]
So,
6cos(36°)sin(18°) = 6 * 1/2[sin(18° + 36°) + sin(18° - 36°)] 6cos(36°)sin(18°) = 3[sin(54°) + sin(-18°)]
Since sin(-x) = -sin(x), we have:
6cos(36°)sin(18°) = 3[sin(54°) - sin(18°)]
Substitute this back into our expression:
3sin(18°) + 6cos(36°)sin(18°) = 3sin(18°) + 3[sin(54°) - sin(18°)] 3sin(18°) + 6cos(36°)sin(18°) = 3sin(18°) + 3sin(54°) - 3sin(18°) 3sin(18°) + 6cos(36°)sin(18°) = 3sin(54°)
Therefore,
12cos(48°)cos(12°)sin(18°) = 3sin(54°)
Knowing sin(18°) and cos(36°)
To proceed, we need to know the values of sin(18°) and cos(36°). These values can be derived using geometric arguments involving a golden triangle, but let's just state them here:
sin(18°) = (√5 - 1) / 4 cos(36°) = (√5 + 1) / 4
Also, we need to find sin(54°). Notice that sin(54°) = cos(90° - 54°) = cos(36°). Therefore,
sin(54°) = (√5 + 1) / 4
Final Calculation
Now we can substitute the value of sin(54°) back into our simplified expression:
3sin(54°) = 3 * (√5 + 1) / 4 3sin(54°) = (3√5 + 3) / 4
So, 12cos(48°)cos(12°)sin(18°) = (3√5 + 3) / 4
Final Answer
Therefore, the exact value of 12cos(48°)cos(12°)sin(18°) is (3√5 + 3) / 4. Looks hard, right? But we solve it simply.
Alternative Approach
Here’s another approach to solve 12cos(48°)cos(12°)sin(18°). We start with the original expression:
12cos(48°)cos(12°)sin(18°)
Multiply and divide by 2:
= 6 * 2cos(48°)cos(12°)sin(18°)
Use the product-to-sum identity 2cos(A)cos(B) = cos(A + B) + cos(A - B):
= 6 * [cos(48° + 12°) + cos(48° - 12°)] * sin(18°) = 6 * [cos(60°) + cos(36°)] * sin(18°)
Since cos(60°) = 1/2:
= 6 * [1/2 + cos(36°)] * sin(18°) = 6sin(18°) * [1/2 + cos(36°)] = 3sin(18°) + 6sin(18°)cos(36°)
Use the product-to-sum identity 2sin(A)cos(B) = sin(A + B) + sin(A - B). So, 6sin(18°)cos(36°) = 3 * 2sin(18°)cos(36°):
= 3 * [sin(18° + 36°) + sin(18° - 36°)] = 3 * [sin(54°) + sin(-18°)]
Since sin(-x) = -sin(x):
= 3 * [sin(54°) - sin(18°)]
Therefore, our original expression becomes:
= 3sin(18°) + 3 * [sin(54°) - sin(18°)] = 3sin(18°) + 3sin(54°) - 3sin(18°) = 3sin(54°)
Now, we know that sin(54°) = cos(90° - 54°) = cos(36°). And cos(36°) = (√5 + 1) / 4.
So:
= 3 * (√5 + 1) / 4 = (3√5 + 3) / 4
Thus, the final answer is:
(3√5 + 3) / 4
Conclusion
So, there you have it, guys! We successfully calculated the value of 12cos(48°)cos(12°)sin(18°) using trigonometric identities and a bit of algebraic manipulation. The key was to use product-to-sum identities to simplify the expression and then recognize the values of sin(18°) and cos(36°). Remember, practice is key when it comes to trigonometry. The more you work with these identities and functions, the easier it will become. Keep exploring, keep practicing, and you'll become a trigonometry master in no time! Great job, and see you in the next math adventure!
