Calculate Tg(β) In A Triangle With Sides 8, 15, And 17

In the realm of trigonometry, understanding the relationships between angles and sides in triangles is fundamental. One such relationship is encapsulated by the tangent function. This article aims to delve into the concept of the tangent of an angle within a right-angled triangle and apply this knowledge to solve a specific problem. We will explore how to calculate the tangent of an angle (${ \beta }$) in a triangle with sides 8, 15, and 17, where ${ \beta }$ is the angle opposite the side of length 15. By the end of this discussion, you will not only be able to determine the value of ${ tg(\beta) }$ but also grasp the underlying trigonometric principles. This comprehensive guide will walk you through the steps, ensuring a clear understanding of the concepts involved. Understanding the tangent function is crucial for various applications in mathematics, physics, and engineering, making this a valuable skill to acquire. Let's embark on this journey to unravel the mysteries of trigonometry and enhance your problem-solving abilities. The following sections will provide a detailed explanation and step-by-step solution to the given problem.

Trigonometric Ratios and the Tangent Function

Before we dive into the specific problem, it's crucial to establish a firm understanding of trigonometric ratios, particularly the tangent. In a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. This fundamental concept forms the bedrock for solving a wide array of trigonometric problems. To truly grasp this, let's break down the key components and their relationships within a right-angled triangle. The longest side, opposite the right angle, is termed the hypotenuse. The other two sides are categorized relative to the angle of interest: the side opposite (the side across from the angle) and the side adjacent (the side next to the angle, not the hypotenuse). The tangent, often abbreviated as 'tan' or 'tg', provides a direct link between these sides and the angle itself. Mathematically, it's represented as: tan(θ) = Opposite / Adjacent. This simple yet powerful formula is the key to unlocking many trigonometric puzzles. By mastering this concept, you'll gain a solid foundation for more advanced topics in trigonometry and its applications. Now, let's apply this knowledge to the given triangle with sides 8, 15, and 17 to determine the tangent of the angle ${ \beta }$.

Identifying the Sides in Our Triangle

In our specific scenario, we have a triangle with sides of lengths 8, 15, and 17. The angle of interest, ${ \beta }$, is opposite the side of length 15. This piece of information is crucial for correctly applying the tangent formula. To begin, we must first confirm that this is indeed a right-angled triangle. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides (a² + b² = c²). In our case, the longest side is 17, so we need to check if 8² + 15² = 17². Calculating this, we have 64 + 225 = 289, which is indeed equal to 17² (289). This confirms that our triangle is a right-angled triangle, and the side with length 17 is the hypotenuse. Now that we've established this, we can confidently identify the other sides relative to angle ${ \beta }$. The side opposite ${ \beta }$ is given as 15, and the side adjacent to ${ \beta }$ is 8. With these values, we are now perfectly positioned to calculate the tangent of ${ \beta }$ using the formula we discussed earlier. This meticulous approach ensures accuracy and a clear understanding of the problem's context. The next step will involve plugging these values into the tangent formula and arriving at our solution.

Calculating tg(β)

Now that we have identified the sides of the triangle and confirmed it's a right-angled triangle, we can proceed with calculating the tangent of angle ${ \beta }$. As we established earlier, the tangent of an angle in a right-angled triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In our case, the side opposite ${ \beta }$ has a length of 15, and the side adjacent to ${ \beta }$ has a length of 8. Therefore, we can express the tangent of ${ \beta }$ as: tg(β) = Opposite / Adjacent = 15 / 8. This simple calculation gives us the value of ${ tg(\beta) }$. It's important to note that the tangent value is a ratio and doesn't have any specific units. It represents the relationship between the lengths of two sides of the triangle. This value can be used in various trigonometric calculations and applications. The tangent function is a fundamental tool in trigonometry, allowing us to relate angles and side lengths in right-angled triangles. By understanding and applying this concept, we can solve a wide range of problems in geometry, physics, and engineering. With this calculation, we have found the value of ${ tg(\beta) }$ in our triangle. The next step is to compare our result with the given answer choices and select the correct option. This process reinforces our understanding and problem-solving skills.

Selecting the Correct Answer

Having calculated that ${ tg(\beta) = \frac{15}{8} }$, we now need to match this result with the provided answer choices. The options given are:

(A) ${ \frac{15}{17} }$ (B) ${ \frac{8}{15} }$ (C) ${ \frac{15}{8} }$ (D) ${ \frac{17}{15} }$

By comparing our calculated value with the options, it becomes clear that option (C), ${ \frac{15}{8} }$, matches our result. Therefore, the correct answer is (C). This step is crucial in any problem-solving process, ensuring that the calculated result is accurately matched with the available choices. It also serves as a final check to confirm the correctness of the solution. In this case, our calculation aligns perfectly with one of the options, reinforcing our confidence in the solution. This exercise demonstrates the importance of understanding the underlying trigonometric principles and applying them systematically to solve problems. The ability to correctly identify and apply trigonometric ratios is a valuable skill in mathematics and its various applications. Now that we have successfully solved the problem and selected the correct answer, let's recap the key steps and concepts involved to solidify our understanding.

Recapitulating the Solution

To ensure a comprehensive understanding, let's recap the steps we took to find the value of ${ tg(\beta) }$ in the triangle with sides 8, 15, and 17. First, we understood the definition of the tangent function as the ratio of the opposite side to the adjacent side in a right-angled triangle. We then confirmed that the given triangle was indeed a right-angled triangle using the Pythagorean theorem. This step was crucial as the tangent function is defined within the context of right-angled triangles. Next, we identified the sides relative to the angle ${ \beta }$. The side opposite ${ \beta }$ was 15, and the side adjacent to ${ \beta }$ was 8. With these values, we applied the tangent formula: tg(β) = Opposite / Adjacent = 15 / 8. This calculation gave us the value of ${ tg(\beta) }$. Finally, we compared our calculated value with the provided answer choices and correctly identified option (C), ${ \frac{15}{8} }$, as the solution. This step-by-step approach highlights the importance of a systematic methodology in problem-solving. By breaking down the problem into smaller, manageable steps, we can ensure accuracy and clarity in our solution. This recapitulation serves as a reinforcement of the concepts and techniques used, solidifying your understanding of trigonometry and its applications. Now, let's delve into some additional insights and practical applications of the tangent function to further enhance your knowledge.

Additional Insights and Applications

Beyond the specific problem we've solved, the tangent function holds significant importance in various fields and applications. Understanding its nuances and practical uses can greatly enhance your problem-solving capabilities in diverse scenarios. In the realm of navigation, the tangent function is used to calculate angles of elevation and depression, which are crucial for determining distances and heights. For instance, if you know the angle of elevation to the top of a building and your distance from the building, you can use the tangent function to calculate the building's height. Similarly, in physics, the tangent function plays a vital role in analyzing projectile motion. The angle at which a projectile is launched affects its range and trajectory, and the tangent function helps in determining these parameters. Engineers also rely heavily on the tangent function in structural design. When designing bridges or buildings, it's essential to calculate the angles and forces involved, and the tangent function is a key tool in these calculations. Furthermore, the tangent function is used in surveying to determine land elevations and distances. Surveyors use instruments to measure angles and then apply trigonometric functions, including the tangent, to calculate distances and heights accurately. These examples illustrate the wide-ranging applicability of the tangent function beyond theoretical mathematics. By grasping its significance and practical uses, you can appreciate its value in solving real-world problems. This broader perspective enriches your understanding and makes you a more versatile problem solver.

In conclusion, we have successfully determined the value of ${ tg(\beta) }$ in the given triangle by applying the principles of trigonometry. This exercise not only provided a solution to a specific problem but also reinforced our understanding of the tangent function and its applications. The journey through this problem, from understanding trigonometric ratios to applying them in a practical scenario, showcases the power of mathematical concepts in problem-solving. By mastering these fundamental principles, you'll be well-equipped to tackle more complex challenges in mathematics and related fields. The key takeaways from this discussion include the definition of the tangent function, its application in right-angled triangles, and the importance of a systematic approach to problem-solving. We hope this comprehensive guide has been insightful and has enhanced your understanding of trigonometry. Keep practicing and exploring the fascinating world of mathematics, and you'll continue to discover its beauty and utility in various aspects of life.