Evaluate: 96√3 Sin(π/48) Cos(π/48) Cos(π/24)...

by Jhon Lennon 48 views

Alright, let's break down this seemingly complex trigonometric expression step-by-step. We're tasked with finding the value of:

96√3 sin(π/48) cos(π/48) cos(π/24) cos(π/12) cos(π/6)

It looks intimidating, but we'll use trigonometric identities to simplify it. Specifically, we’ll repeatedly apply the double-angle formula for sine, which states:

sin(2θ) = 2sin(θ)cos(θ)

This identity allows us to combine sine and cosine terms, effectively doubling the angle inside the sine function at each step. Let's dive into the step-by-step evaluation.

Step-by-Step Evaluation

Step 1: Apply the Double-Angle Formula to the First Two Terms

We have sin(π/48) and cos(π/48). Let's pair them up using the double-angle formula. To do this, we'll multiply and divide by 2:

96√3 sin(π/48) cos(π/48) cos(π/24) cos(π/12) cos(π/6) = 96√3 * (1/2) * [2sin(π/48) cos(π/48)] cos(π/24) cos(π/12) cos(π/6)

Now, we can apply the double-angle formula:

2sin(π/48) cos(π/48) = sin(2 * π/48) = sin(π/24)

So our expression becomes:

96√3 * (1/2) * sin(π/24) cos(π/24) cos(π/12) cos(π/6) = 48√3 sin(π/24) cos(π/24) cos(π/12) cos(π/6)

Step 2: Apply the Double-Angle Formula Again

We now have sin(π/24) and cos(π/24). Again, we multiply and divide by 2:

48√3 sin(π/24) cos(π/24) cos(π/12) cos(π/6) = 48√3 * (1/2) * [2sin(π/24) cos(π/24)] cos(π/12) cos(π/6)

Applying the double-angle formula:

2sin(π/24) cos(π/24) = sin(2 * π/24) = sin(π/12)

Our expression is now:

48√3 * (1/2) * sin(π/12) cos(π/12) cos(π/6) = 24√3 sin(π/12) cos(π/12) cos(π/6)

Step 3: Apply the Double-Angle Formula Once More

We have sin(π/12) and cos(π/12). Multiply and divide by 2:

24√3 sin(π/12) cos(π/12) cos(π/6) = 24√3 * (1/2) * [2sin(π/12) cos(π/12)] cos(π/6)

Applying the double-angle formula:

2sin(π/12) cos(π/12) = sin(2 * π/12) = sin(π/6)

Our expression simplifies to:

24√3 * (1/2) * sin(π/6) cos(π/6) = 12√3 sin(π/6) cos(π/6)

Step 4: Apply the Double-Angle Formula for the Final Time

We've got sin(π/6) and cos(π/6). One last time, multiply and divide by 2:

12√3 sin(π/6) cos(π/6) = 12√3 * (1/2) * [2sin(π/6) cos(π/6)]

Applying the double-angle formula:

2sin(π/6) cos(π/6) = sin(2 * π/6) = sin(π/3)

So our expression is now:

12√3 * (1/2) * sin(π/3) = 6√3 sin(π/3)

Step 5: Evaluate sin(π/3)

We know that sin(π/3) = sin(60°) = √3/2. Substitute this value into our expression:

6√3 sin(π/3) = 6√3 * (√3/2)

Step 6: Simplify to Get the Final Result

6√3 * (√3/2) = 6 * (3/2) = 18/2 = 9

Therefore, the value of the expression is 9.

Summary of Steps

To reiterate, here's what we did:

  1. Repeatedly used the identity sin(2θ) = 2sin(θ)cos(θ).
  2. Halved the constant term (96√3) successively as we applied the identity.
  3. Doubled the angle inside the sine function at each step, going from π/48 to π/24, then to π/12, then to π/6, and finally to π/3.
  4. Evaluated sin(π/3) as √3/2.
  5. Simplified the expression to arrive at the final answer of 9.

Why This Works: The Power of the Double-Angle Formula

The double-angle formula is a cornerstone of trigonometry. It allows us to relate trigonometric functions of an angle to those of twice that angle. In this problem, its repeated application was key to simplifying the expression. Each time we applied it, we effectively "absorbed" a cosine term into the sine term, while simultaneously doubling the angle. This process continued until we were left with a single sine term with a manageable angle (π/3).

Think of it like peeling an onion, where each layer (application of the formula) reveals a simpler core. Without this identity, directly calculating the value of sin(π/48) and cos(π/48) would be far more complex and likely require numerical approximations.

Alternative Approaches (and Why They're Less Ideal)

While it's theoretically possible to calculate the values of sin(π/48), cos(π/48), cos(π/24), cos(π/12), and cos(π/6) individually and then multiply them all together, this approach is highly impractical for a few reasons:

  • Complexity: Calculating sin(π/48) and cos(π/48) directly would involve nested radicals or the use of half-angle formulas multiple times, leading to a very messy expression.
  • Error Accumulation: Each individual calculation would introduce a potential for rounding errors, and these errors would compound when multiplying all the values together, leading to a less accurate final result.
  • Lack of Elegance: This brute-force method misses the beautiful simplification that the double-angle formula provides. Mathematics is often about finding the most efficient and elegant solution.

Therefore, the repeated application of the double-angle formula is by far the most efficient and accurate way to evaluate this expression. It transforms a daunting problem into a series of straightforward steps, showcasing the power and elegance of trigonometric identities.

Key Takeaways

  • The double-angle formula is a powerful tool for simplifying trigonometric expressions.
  • Repeated application of trigonometric identities can often lead to significant simplifications.
  • Look for opportunities to use identities to avoid direct, complex calculations.
  • Efficiency and elegance are often hallmarks of a good mathematical solution.

So, next time you encounter a trigonometric expression that seems overwhelming, remember the double-angle formula and see if you can "peel away" the layers to reveal a simpler solution! It might just save you a lot of time and effort, and you'll feel pretty clever in the process. Keep exploring and practicing, and you'll become a trigonometric wizard in no time!