Evaluate Cos(48)cos(12) * 38730 / (5 * 8)

by Jhon Lennon 42 views

Let's dive into evaluating the expression cos(48°)cos(12°) * 38730 / (5 * 8). This problem involves trigonometric functions and arithmetic operations, so we'll break it down step by step to make sure we get to the correct answer. Guys, get ready for some math magic!

Understanding the Trigonometric Part

First, we need to understand and simplify the trigonometric part of the expression: cos(48°)cos(12°). We can use trigonometric identities to help us simplify this product. A common identity that comes in handy when dealing with products of cosine functions is the product-to-sum identity. Specifically, we can use the identity:

cos(A)cos(B) = 0.5 * [cos(A - B) + cos(A + B)]

In our case, A = 48° and B = 12°. Plugging these values into the identity, we get:

cos(48°)cos(12°) = 0.5 * [cos(48° - 12°) + cos(48° + 12°)] cos(48°)cos(12°) = 0.5 * [cos(36°) + cos(60°)]

Now, we know that cos(60°) = 0.5. The value of cos(36°) is a bit less common, but it's a known value that can be expressed using the golden ratio. Specifically:

cos(36°) = (1 + √5) / 4

So, we can substitute these values back into our expression:

cos(48°)cos(12°) = 0.5 * [(1 + √5) / 4 + 0.5] cos(48°)cos(12°) = 0.5 * [(1 + √5) / 4 + 2 / 4] cos(48°)cos(12°) = 0.5 * [(3 + √5) / 4] cos(48°)cos(12°) = (3 + √5) / 8

Simplifying the Arithmetic Part

Next, let's simplify the arithmetic part of the expression: 38730 / (5 * 8). This is straightforward:

38730 / (5 * 8) = 38730 / 40 38730 / 40 = 968.25

So, the arithmetic part simplifies to 968.25.

Combining the Parts

Now, we combine the trigonometric and arithmetic parts:

cos(48°)cos(12°) * 38730 / (5 * 8) = [(3 + √5) / 8] * 968.25

We know that √5 ≈ 2.236, so:

(3 + √5) / 8 ≈ (3 + 2.236) / 8 (3 + √5) / 8 ≈ 5.236 / 8 (3 + √5) / 8 ≈ 0.6545

Now, multiply this by 968.25:

0.6545 * 968.25 ≈ 633.71

Therefore, the value of the expression is approximately 633.71.

Detailed Calculation

Let's do a more precise calculation to get a better estimate. We have:

cos(48°)cos(12°) * 38730 / (5 * 8) = [(3 + √5) / 8] * (38730 / 40)

We already found that:

cos(48°)cos(12°) = (3 + √5) / 8

And:

38730 / 40 = 968.25

So, we need to calculate:

[(3 + √5) / 8] * 968.25

Using a calculator, we find that:

√5 ≈ 2.2360679775

So:

(3 + √5) / 8 ≈ (3 + 2.2360679775) / 8 (3 + √5) / 8 ≈ 5.2360679775 / 8 (3 + √5) / 8 ≈ 0.65450849718

Now, multiply this by 968.25:

0. 65450849718 * 968.25 ≈ 633.7086646

Rounding to two decimal places, we get approximately 633.71.

Final Answer

So, after breaking down the expression and calculating each part carefully, we find that:

cos(48°)cos(12°) * 38730 / (5 * 8) ≈ 633.71

This detailed step-by-step approach ensures that we've addressed each component accurately, providing a comprehensive solution to the problem. It's always a good idea to double-check each step to minimize errors and ensure the most accurate result. Remember, math can be fun when you break it down into manageable parts! Keep practicing, and you'll become a math whiz in no time!

Alternative Approach using Calculator

For those who prefer a more direct approach, using a calculator can significantly speed up the process, especially when dealing with trigonometric functions. Here's how you can solve the problem using a calculator:

  1. Calculate cos(48°): Make sure your calculator is in degree mode. Input cos(48) and you should get approximately 0.66913060635. Store this value or write it down.
  2. Calculate cos(12°): Similarly, input cos(12) and you should get approximately 0.97814760073. Store this value or write it down.
  3. Multiply the cosine values: Multiply the two values you obtained: 0.66913060635 * 0.97814760073 ≈ 0.65450849718.
  4. Calculate 38730 / (5 * 8): This is the arithmetic part. 5 * 8 = 40, so you have 38730 / 40 = 968.25.
  5. Multiply the results: Now, multiply the result from step 3 by the result from step 4: 0.65450849718 * 968.25 ≈ 633.7086646.

Rounding to two decimal places, you get approximately 633.71. This method provides a quick and efficient way to solve the problem, especially when accuracy is crucial.

Advantages of using a calculator:

  • Speed: Calculators perform calculations much faster than manual methods.
  • Accuracy: Calculators provide accurate results, reducing the risk of human error.
  • Convenience: Calculators are readily available and easy to use.

Disadvantages of using a calculator:

  • Dependence: Over-reliance on calculators can hinder understanding of underlying mathematical principles.
  • Battery/Power: Calculators require power, which may not always be available.
  • Potential for errors: Incorrect input can lead to incorrect results.

Using a calculator is a great tool for solving complex expressions quickly and accurately. Just ensure you understand the steps and the principles involved to avoid relying solely on the device.