Finding Side X In A Right-Angled Triangle Trigonometry Problem Solved
Finding the unknown side in a right-angled triangle is a fundamental concept in trigonometry. In this article, we'll explore a specific problem where we need to determine the value of 'x' in an upside-down right-angled triangle. We'll be given the hypotenuse (4cm) and one of the angles (35°) and will use trigonometric principles to calculate the length of the side adjacent to the given angle. This problem will illustrate how the cosine function can be applied to solve for unknown lengths in triangles, a crucial skill in various fields of mathematics, physics, and engineering.
Understanding the Problem
Before we dive into the solution, let's clarify the problem. We have an upside-down right-angled triangle, meaning the right angle (90°) is at the bottom. The hypotenuse, which is the side opposite the right angle, measures 4cm. The angle at the top of the triangle is 35°. Our goal is to find the length of the side adjacent to the 35° angle, which we'll call 'x'.
To effectively solve this problem, it's crucial to have a firm grasp of trigonometric ratios. These ratios relate the angles of a right-angled triangle to the lengths of its sides. The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). Each of these ratios is defined as the relationship between two sides of the triangle concerning a specific angle. For instance, the sine of an angle is the ratio of the opposite side to the hypotenuse, the cosine of an angle is the ratio of the adjacent side to the hypotenuse, and the tangent of an angle is the ratio of the opposite side to the adjacent side. Understanding these ratios is paramount in tackling problems involving triangles and angles.
In our case, we are given the hypotenuse and need to find the adjacent side. This means the cosine function is the most suitable trigonometric ratio to apply. The cosine of an angle in a right-angled triangle is defined as the length of the adjacent side divided by the length of the hypotenuse. By understanding this relationship, we can set up an equation that will allow us to solve for the unknown length, 'x'. This approach highlights the power of trigonometry in bridging the gap between angles and side lengths, enabling us to solve geometric problems using algebraic methods. So, with a clear understanding of the problem and the relevant trigonometric ratio, we are now ready to proceed with the solution.
Applying the Cosine Function
The key to solving this problem lies in applying the cosine function. As mentioned earlier, the cosine of an angle in a right-angled triangle is defined as the ratio of the adjacent side to the hypotenuse. Mathematically, this is expressed as:
cos(θ) = Adjacent / Hypotenuse
In our scenario, θ (theta) represents the given angle, which is 35°. The hypotenuse is given as 4cm, and the side adjacent to the 35° angle is what we need to find, which we've labeled as 'x'. By substituting these values into the cosine formula, we get:
cos(35°) = x / 4
Now, to isolate 'x' and solve for its value, we need to multiply both sides of the equation by 4. This gives us:
x = 4 * cos(35°)
At this point, we can use a calculator to find the cosine of 35 degrees. Make sure your calculator is set to degree mode, as we're dealing with angles measured in degrees. The cosine of 35° is approximately 0.819.
Substituting this value back into our equation, we have:
x = 4 * 0.819
Now, we simply perform the multiplication to find the value of x. This step demonstrates the practical application of trigonometric ratios in real-world scenarios. By using the cosine function, we've transformed a geometric problem into a simple algebraic equation that we can easily solve. The ability to make this transformation is a cornerstone of trigonometry and is essential for solving a wide range of problems in various fields. The next step involves the actual calculation and rounding off the answer to the required decimal place, which will give us the final value of 'x'.
Calculating the Value of x
Having set up the equation x = 4 * cos(35°), and having found that cos(35°) ≈ 0.819, we can now calculate the value of x:
x = 4 * 0.819 = 3.276
The problem asks for the answer to be corrected to one decimal place. To do this, we look at the second decimal place in our result, which is 7. Since 7 is greater than or equal to 5, we round up the first decimal place. Therefore, 3.276 rounded to one decimal place is 3.3.
So, the value of x, correct to one decimal place, is 3.3 cm. This final calculation is a crucial step in solving the problem. It demonstrates the importance of not only understanding the trigonometric principles but also being able to perform the necessary arithmetic operations accurately. Rounding off the answer to the specified decimal place ensures that the solution is presented in the required format and level of precision. This attention to detail is essential in mathematical problem-solving and reflects a thorough understanding of the problem's requirements. With this final step, we have successfully found the value of 'x' in the given right-angled triangle problem.
Solution
Based on our calculations, the value of x in the upside-down right-angled triangle, corrected to one decimal place, is 3.3 cm. Therefore, the correct answer is option A, which is 3.3cm.
In summary, we approached this problem by first understanding the given information: the hypotenuse (4cm) and one angle (35°) of a right-angled triangle. We identified that we needed to find the length of the side adjacent to the given angle. Recognizing that the cosine function relates the adjacent side, hypotenuse, and angle, we applied the formula cos(θ) = Adjacent / Hypotenuse. We then substituted the given values into the formula, resulting in the equation cos(35°) = x / 4. By multiplying both sides by 4, we isolated x and obtained x = 4 * cos(35°). Using a calculator, we found that cos(35°) ≈ 0.819, and thus, x ≈ 4 * 0.819 = 3.276. Finally, we rounded this result to one decimal place, giving us x = 3.3 cm.
This solution highlights the step-by-step process of solving trigonometric problems. It emphasizes the importance of understanding the problem, identifying the relevant trigonometric ratio, applying the formula correctly, performing the calculations accurately, and presenting the answer in the required format. This systematic approach is crucial for success in mathematics and related fields. By breaking down the problem into manageable steps and applying the principles of trigonometry, we were able to confidently arrive at the correct solution.
Conclusion
In this article, we successfully found the value of x in a given right-angled triangle problem. We utilized the cosine function, a fundamental concept in trigonometry, to relate the angle and sides of the triangle. By following a step-by-step approach, we were able to calculate the value of x to be 3.3 cm, correct to one decimal place.
This exercise underscores the importance of trigonometry in solving geometric problems. The ability to apply trigonometric ratios allows us to bridge the gap between angles and side lengths, enabling us to find unknown values in triangles. This skill is not only essential in mathematics but also has wide-ranging applications in fields such as physics, engineering, and surveying. The systematic approach we used, which involves understanding the problem, identifying the relevant trigonometric ratio, applying the formula, performing calculations, and rounding off the answer, is a valuable problem-solving strategy that can be applied to various mathematical challenges.
The problem we solved is a classic example of how trigonometric principles can be used in real-world scenarios. Whether it's calculating the height of a building, determining the distance across a river, or navigating an aircraft, trigonometry provides the tools and techniques needed to solve these problems accurately. By mastering these principles, students and professionals alike can gain a deeper understanding of the world around them and develop the problem-solving skills necessary to tackle complex challenges. The value of 'x' we found in this exercise serves as a concrete example of the power and practicality of trigonometry.
By working through this problem, we've reinforced our understanding of trigonometric ratios and their applications. We've also honed our problem-solving skills and developed a systematic approach to tackling mathematical challenges. These are valuable skills that will serve us well in future mathematical endeavors and beyond.
