Finding Sin(E) In Right Triangle DEF A Step-by-Step Solution

In the realm of mathematics, particularly trigonometry, understanding the relationships between angles and sides in a right triangle is fundamental. This article delves into a specific problem involving a right triangle, focusing on calculating the sine of an angle. We'll break down the problem step-by-step, ensuring a clear and comprehensive understanding of the solution. The problem presents us with a right triangle DEF, where angle EDF is the right angle. The length of side DE is given as 8 units, and the length of the hypotenuse EF is 10 units. Our task is to determine the value of sin(E). This involves applying the basic trigonometric ratios within the context of a right triangle, a concept crucial for various fields including engineering, physics, and computer graphics. Trigonometry, at its core, is the study of the relationships between angles and sides of triangles. The three primary trigonometric ratios – sine, cosine, and tangent – form the bedrock of this field. These ratios relate the angles of a right triangle to the lengths of its sides. The sine of an angle, often abbreviated as sin, is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine of an angle, abbreviated as cos, is the ratio of the length of the adjacent side to the hypotenuse. And finally, the tangent of an angle, abbreviated as tan, is the ratio of the opposite side to the adjacent side. These ratios are essential tools for solving a wide range of problems involving triangles and angles.

Decoding the Right Triangle DEF

To effectively calculate sin(E), we need to visualize and understand the given right triangle DEF. The triangle is defined by three vertices: D, E, and F. The crucial piece of information is that angle EDF is a right angle, which means it measures 90 degrees. This right angle is essential because it allows us to apply the trigonometric ratios directly. In a right triangle, the side opposite the right angle is called the hypotenuse, which is the longest side of the triangle. In our case, the hypotenuse is EF, with a given length of 10 units. The other two sides, DE and DF, are called legs. Side DE is given as 8 units long. To find sin(E), we need to determine the length of the side opposite angle E. Looking at the triangle, the side opposite angle E is DF. However, the length of DF is not directly provided. This is where the Pythagorean theorem comes into play. The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is expressed as: a² + b² = c², where 'a' and 'b' are the lengths of the legs, and 'c' is the length of the hypotenuse. Applying this theorem to our triangle DEF, we can find the length of DF.

Applying the Pythagorean Theorem

Now, let's put the Pythagorean theorem into action to find the missing side DF. We know that DE = 8 and EF (the hypotenuse) = 10. Let's denote the length of DF as 'x'. According to the Pythagorean theorem, we have: DE² + DF² = EF². Substituting the known values, we get: 8² + x² = 10². This simplifies to: 64 + x² = 100. To isolate x², we subtract 64 from both sides of the equation: x² = 100 - 64, which gives us x² = 36. Now, to find the value of x, we take the square root of both sides: x = √36. Therefore, x = 6. This tells us that the length of side DF is 6 units. With the length of DF now known, we have all the necessary information to calculate sin(E). We know that sin(E) is defined as the ratio of the length of the side opposite angle E (which is DF) to the length of the hypotenuse (which is EF). So, sin(E) = DF / EF. Plugging in the values we found, we have sin(E) = 6 / 10. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So, 6 / 10 simplifies to 3 / 5. Therefore, the value of sin(E) is 3/5.

Calculating sin(E) and Identifying the Correct Option

With the length of side DF determined to be 6 units, we can now precisely calculate the value of sin(E). Remember, the sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. In triangle DEF, the side opposite angle E is DF, and the hypotenuse is EF. We have already established that DF = 6 and EF = 10. Therefore, sin(E) = DF / EF = 6 / 10. To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 2. This gives us sin(E) = (6 ÷ 2) / (10 ÷ 2) = 3 / 5. Now, let's examine the options provided in the original problem. The options were: Three-fifths, Three-fourths, Four-fifths, and Four-thirds. Comparing our calculated value of sin(E) with the options, we can clearly see that the correct option is Three-fifths. This result aligns perfectly with our calculations and understanding of trigonometric ratios in right triangles. This exercise highlights the importance of the Pythagorean theorem in finding missing side lengths in right triangles, which then allows us to calculate trigonometric ratios like sine, cosine, and tangent.

Conclusion: The Significance of Trigonometric Ratios

In conclusion, by carefully applying the Pythagorean theorem and the definition of the sine function, we successfully determined that the value of sin(E) in right triangle DEF is 3/5. This exercise underscores the fundamental role of trigonometric ratios in solving geometric problems. Understanding the relationships between angles and sides in right triangles is not just a mathematical concept; it has profound implications in various fields. From architecture and engineering, where precise angle and length calculations are crucial for structural stability, to navigation and surveying, where trigonometric principles are used to determine distances and positions, these ratios are indispensable tools. Furthermore, in computer graphics and game development, trigonometric functions are used extensively for rotations, projections, and other transformations. The ability to calculate sine, cosine, and tangent, along with a solid grasp of the Pythagorean theorem, provides a powerful foundation for tackling a wide range of real-world problems. The problem-solving process we've walked through here, from identifying the key information to applying the appropriate theorems and definitions, is a valuable skill that extends far beyond the realm of mathematics. It encourages logical thinking, attention to detail, and the ability to break down complex problems into manageable steps. By mastering these skills, we not only enhance our mathematical abilities but also develop a crucial toolkit for success in various aspects of life.

This exploration of triangle DEF and the calculation of sin(E) serves as a microcosm of the broader applications of trigonometry. The principles we've discussed are the building blocks for more advanced concepts and techniques, demonstrating the enduring relevance and importance of trigonometry in our world.