Solving Tan(6°) = F/2.3: A Step-by-Step Guide

Let's dive into how to solve the trigonometric equation ${\tan 6^{\circ} = \frac{f}{2.3}}$. This problem involves using trigonometry to find the unknown variable, f, and then rounding our answer to two decimal places. This is a common type of problem in mathematics, particularly in trigonometry and algebra. Understanding how to tackle this kind of problem is crucial for anyone studying these subjects, as it combines trigonometric functions with basic algebraic manipulation. So, let's break it down step by step to ensure we understand the process thoroughly.

Understanding the Trigonometric Equation

To solve trigonometric equations like ${\tan 6^{\circ} = \frac{f}{2.3}}$, you first need to understand what each part of the equation represents. The ${\tan}$ function, short for tangent, is a trigonometric function that relates an angle of a right-angled triangle to the ratio of the opposite side to the adjacent side. In this case, ${\tan 6^{\circ}}$ refers to the tangent of a 6-degree angle. This is a specific numerical value that can be found using a calculator or trigonometric tables. The variable f represents the unknown value we are trying to find. The number 2.3 is a known constant. Our goal is to isolate f on one side of the equation to find its value. This involves using algebraic principles to manipulate the equation until f is by itself, allowing us to calculate its numerical value. Understanding these components is essential for solving not just this equation, but any trigonometric equation you might encounter.

Isolating the Variable f

Now, let's focus on isolating the variable f in the equation ${\tan 6^{\circ} = \frac{f}{2.3}}$. The key to isolating a variable is to perform operations on both sides of the equation in a way that cancels out everything else surrounding the variable. In this case, f is being divided by 2.3. To undo this division, we need to multiply both sides of the equation by 2.3. This is a fundamental algebraic principle: whatever operation you perform on one side of an equation, you must also perform on the other side to maintain the equality. So, we multiply both ${\tan 6^{\circ}}$ and ${\frac{f}{2.3}}$ by 2.3. This step is crucial because it simplifies the equation, bringing us closer to finding the value of f. After performing this multiplication, the equation will be in a form where f is alone on one side, and we can then calculate its value.

Calculating tan(6°)

The next step in solving the equation ${\tan 6^{\circ} = \frac{f}{2.3}}$ is to calculate the value of tan(6°). This is where a calculator comes in handy. Make sure your calculator is set to degree mode, as the angle is given in degrees. If the calculator is in radian mode, the result will be incorrect. Inputting ${\tan 6^{\circ}}$ into your calculator will give you a decimal value. It’s important to note that this value is an approximation, as the tangent of most angles is an irrational number, meaning its decimal representation goes on infinitely without repeating. Therefore, we typically round this value to a certain number of decimal places for practical purposes. Once we have the decimal value of ${\tan 6^{\circ}}$, we can substitute it back into our equation. This step transforms the trigonometric part of the equation into a simple arithmetic problem, making it easier to solve for f. The accuracy of our final answer depends on how accurately we calculate ${\tan 6^{\circ}}$, so using a calculator correctly is essential.

Performing the Calculation

After calculating ${\tan 6^{\circ}}$ and isolating f, the next step is to perform the final calculation to find the value of f. From the previous steps, we have the equation ${f = 2.3 imes \tan 6^{\circ}}$. We've already found the value of ${\tan 6^{\circ}}$ using a calculator, so now we simply need to multiply this value by 2.3. This multiplication will give us the numerical value of f. It’s crucial to pay attention to the order of operations and ensure that the multiplication is carried out correctly. This step is a straightforward arithmetic operation, but accuracy is key. A small error in this multiplication can lead to an incorrect final answer. Once we have the result of this multiplication, we have found the value of f.

Rounding the Answer to 2 Decimal Places

Rounding the answer to 2 decimal places is the final step in solving the equation ${\tan 6^{\circ} = \frac{f}{2.3}}$. In many practical and academic contexts, it’s important to provide answers to a specified level of precision. Rounding to two decimal places means we want to keep only two digits after the decimal point. To do this, we look at the third digit after the decimal point. If this digit is 5 or greater, we round up the second digit. If it’s less than 5, we leave the second digit as it is. This process ensures that our answer is accurate to the nearest hundredth. Rounding is not just about simplifying the answer; it’s also about presenting it in a way that is appropriate for the context of the problem. In many real-world applications, providing an answer with too many decimal places can be impractical or even misleading. Therefore, understanding how to round correctly is an essential skill in mathematics and beyond.

Detailed Solution

Here’s a detailed solution to the equation ${\tan 6^{\circ} = \frac{f}{2.3}}$, broken down step by step:

1. Isolate f: To isolate f, we multiply both sides of the equation by 2.3:

   ${ 2. 3 imes \tan 6^{\circ} = f }$

2. Calculate tan(6°): Using a calculator, we find that:

   ${ \tan 6^{\circ} \approx 0.1051 }$

3. Substitute and calculate f: Now, substitute the value of ${\tan 6^{\circ}}$ into the equation:

   ${ f = 2.3 imes 0.1051 }$

   ${ f \approx 0.2417 }$

4. Round to 2 decimal places: Rounding 0.2417 to two decimal places gives us:

   ${ f \approx 0.24 }$

Therefore, the solution to the equation ${\tan 6^{\circ} = \frac{f}{2.3}}$, rounded to 2 decimal places, is approximately 0.24.

Conclusion

In conclusion, solving the equation ${\tan 6^{\circ} = \frac{f}{2.3}}$ involves understanding trigonometric functions, algebraic manipulation, and the importance of precision in calculations. We started by isolating the variable f, then calculated the value of ${\tan 6^{\circ}}$ using a calculator, performed the multiplication, and finally rounded our answer to two decimal places. This step-by-step process not only helps us find the solution but also reinforces key mathematical concepts. Remember, accuracy and attention to detail are crucial in mathematics, as small errors can lead to incorrect results. By following these steps carefully, you can confidently tackle similar trigonometric problems. For further reading and a deeper understanding of trigonometry, you might find valuable resources at Khan Academy's Trigonometry section.