Finding The Length Of AB Using Trigonometry

In trigonometry, solving for unknown lengths in triangles often involves using trigonometric ratios. In this article, we will explore the given equation $\sin ({25}^{\circ })=\frac{9}{c}$\sin(25^{\circ}) = \frac{9}{c}} to find the length of ${\overline{AB}$, which corresponds to the variable c in the equation. This problem highlights the application of the sine function in determining side lengths in right triangles. We will go through the steps to solve for c, discuss the underlying trigonometric principles, and ensure the final answer is rounded to the nearest tenth.

Understanding the Trigonometric Context

To accurately determine the length of ${\overline{AB}}$, we must first understand the fundamental trigonometric principles at play. The equation ${\sin(25^{\circ}) = \frac{9}{c}}$ suggests we are dealing with a right triangle where the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this context, 25 degrees is the angle, 9 represents the length of the side opposite this angle, and c represents the hypotenuse, which is the side we are trying to find. The sine function, denoted as sin(${\theta}$), is one of the primary trigonometric ratios, alongside cosine (cos) and tangent (tan). These ratios are used to relate the angles of a triangle to the lengths of its sides. In a right-angled triangle, the sine of an angle is specifically the ratio of the length of the opposite side to the hypotenuse. Therefore, the equation ${\sin(25^{\circ}) = \frac{9}{c}}$ is a direct application of this principle, allowing us to set up a proportion that relates the sine of the angle to the side lengths. To solve for c, we will need to manipulate this equation algebraically, isolating c on one side. This process involves using the properties of equations to perform the same operations on both sides, ensuring the equation remains balanced. The accurate determination of c requires a clear understanding of these trigonometric and algebraic principles, making this problem a comprehensive exercise in mathematical problem-solving. We will also discuss how to utilize a calculator effectively to find the sine of an angle and how to correctly round the final answer to the nearest tenth, as specified in the problem. This detailed approach ensures a thorough understanding and accurate solution to the problem at hand, providing a solid foundation for future trigonometric applications.

Solving for ${c}$

The core of solving for the length ${c}$ in the equation ${\sin(25^{\circ}) = \frac{9}{c}}$ involves a series of algebraic manipulations to isolate the variable c. The initial equation presents c in the denominator, making it necessary to rearrange the equation to solve for c directly. The first step in this process is to multiply both sides of the equation by c. This operation effectively removes c from the denominator on the right side, resulting in the equation ${c \cdot \sin(25^{\circ}) = 9}$. By performing this multiplication, we maintain the balance of the equation while bringing c to the numerator. The next step is to isolate c by dividing both sides of the equation by ${\sin(25^{\circ})}$. This will give us the equation ${c = \frac{9}{\sin(25^{\circ})}}$, which expresses c in terms of a known value (9) and the sine of a known angle (25 degrees). Now that we have isolated c, we can proceed to calculate its value. This involves finding the sine of 25 degrees, which can be done using a calculator. It is crucial to ensure the calculator is in degree mode for this calculation to be accurate. The sine of 25 degrees is approximately 0.4226. Substituting this value into our equation gives us ${c = \frac{9}{0.4226}}$. Performing this division yields a value for c that represents the length of ${\overline{AB}}$. However, this value is an approximation, and to provide the answer in the requested format, we need to round it to the nearest tenth. This rounding process involves looking at the digit in the hundredths place to determine whether to round the tenths digit up or leave it as is. This step-by-step approach ensures accuracy and clarity in solving for c, highlighting the importance of algebraic manipulation and the use of trigonometric functions.

Calculating the Value of ${c}$ and Rounding

Calculating the value of ${c}$ requires a precise approach, especially when rounding to the nearest tenth. From the previous steps, we have the equation ${c = \frac{9}{\sin(25^{\circ})}}$. We've already established that ${\sin(25^{\circ})}$ is approximately 0.4226. Therefore, the next step is to divide 9 by 0.4226. Using a calculator, ${\frac{9}{0.4226}}$ gives us approximately 21.30. This value is a decimal number, and to meet the requirement of rounding to the nearest tenth, we need to consider the digit in the hundredths place. In this case, the hundredths digit is 0. According to the rules of rounding, if the hundredths digit is 5 or greater, we round up the tenths digit. If it is less than 5, we leave the tenths digit as it is. Since 0 is less than 5, we do not round up the tenths digit. Therefore, the value of c rounded to the nearest tenth is 21.3. This process of calculating and rounding is critical in many mathematical and scientific applications. It ensures that the final answer is both accurate and presented in the required format. The careful use of a calculator and adherence to rounding rules are essential skills in problem-solving. By understanding these steps, one can confidently solve similar problems involving trigonometric ratios and rounding. Moreover, this detailed approach helps in avoiding common errors and ensures the solution is both mathematically sound and practically useful. The result, 21.3, represents the approximate length of ${\overline{AB}}$ in the units implied by the problem, which are inches in this context.

Selecting the Correct Answer

After calculating the value of ${c}$ and rounding it to the nearest tenth, we arrive at 21.3. Now, we need to select the correct answer from the given options. The options are:

A. 19.3 in B. 21.3 in C. 23.5 in D. 68.0 in

Comparing our calculated value (21.3) with the options, we can see that option B, 21.3 in, matches our result. This confirms that 21.3 inches is the length of ${\overline{AB}}$, rounded to the nearest tenth. The other options, 19.3 in, 23.5 in, and 68.0 in, do not match our calculated value. Therefore, they are incorrect. Selecting the correct answer is a critical step in problem-solving, as it ensures that the final result is accurately communicated. In this case, the process of elimination further solidifies the correctness of option B. By methodically working through the problem, from understanding the trigonometric context to performing the calculations and rounding, we can confidently arrive at the correct solution. This approach demonstrates the importance of careful analysis and attention to detail in mathematical problem-solving. The ability to accurately calculate and compare values with given options is a fundamental skill that is applicable in various fields beyond mathematics, making this exercise a valuable learning experience.

Conclusion

In conclusion, by applying trigonometric principles and algebraic manipulation, we successfully determined the length of ${\overline{AB}}$ using the given equation ${\sin(25^{\circ}) = \frac{9}{c}}$. The steps involved understanding the sine function, rearranging the equation to solve for c, calculating the value of ${\sin(25^{\circ})}$, performing the division, and rounding the final answer to the nearest tenth. This process led us to the solution ${c \approx 21.3}$ inches, which corresponds to option B in the given choices. This problem exemplifies the practical application of trigonometry in solving real-world problems involving triangles. The ability to accurately use trigonometric ratios and algebraic techniques is essential in various fields, including engineering, physics, and architecture. Moreover, this exercise reinforces the importance of careful calculation, attention to detail, and the proper use of mathematical tools such as calculators. The systematic approach demonstrated here—from understanding the problem to selecting the correct answer—highlights the core principles of effective problem-solving. By mastering these skills, students and professionals alike can confidently tackle complex challenges and achieve accurate and reliable results. The final answer, 21.3 inches, not only solves the specific problem at hand but also provides a valuable learning experience that can be applied to a wide range of similar scenarios.