Given A Rectangle ABCD With A Perimeter Of 10 And A Shorter Side BC Of Length 2, Calculate The Trigonometric Function Values Of The Acute Angles Of Triangle ABC.

Jun 20, 2025 by ADMIN 162 views

Calculating Trigonometric Functions in a Rectangle: A Step-by-Step Guide

In this article, we will explore a classic geometry problem involving a rectangle and its trigonometric functions. We are given a rectangle ABCD with a perimeter of 10 and a shorter side BC of length 2. Our goal is to calculate the values of the trigonometric functions for the acute angles of triangle ABC. This problem combines basic geometric principles with trigonometric concepts, offering a comprehensive exercise for students and enthusiasts alike. Let's delve into the solution step by step.

Understanding the Problem: Rectangle ABCD and Triangle ABC

To effectively solve this problem, it's crucial to first understand the given information and visualize the geometric shapes involved. We have a rectangle, a fundamental quadrilateral with specific properties, and a triangle formed within it. Rectangles, by definition, are quadrilaterals with four right angles, meaning each corner forms a 90-degree angle. This property is essential because it allows us to apply the Pythagorean theorem and trigonometric ratios with confidence. In our case, rectangle ABCD has sides AB, BC, CD, and DA. We know that the opposite sides of a rectangle are equal in length, and all its angles are right angles. The perimeter of a rectangle, which is the total length of all its sides, is given as 10. This information is critical for determining the lengths of the unknown sides.

The shorter side BC is given as 2. Since opposite sides of a rectangle are equal, we also know that side DA is 2. This leaves us to determine the lengths of sides AB and CD, which are also equal. To find these lengths, we'll use the perimeter formula. Let's denote the length of side AB (and CD) as 'x'. The perimeter of the rectangle is then 2 * (AB + BC) or 2 * (x + 2). We know this perimeter is 10, so we can set up the equation 2 * (x + 2) = 10. Solving for x will give us the length of the longer side of the rectangle. Once we find the length of AB, we will have all the information needed to describe rectangle ABCD fully.

The next important element is triangle ABC, which is formed by the vertices A, B, and C of the rectangle. Triangle ABC is a right-angled triangle because angle B in the rectangle is a right angle. This is a key observation because it means we can directly apply trigonometric ratios to the acute angles of this triangle. The sides of triangle ABC are AB, BC, and AC. We already know the lengths of AB and BC, and we can find the length of AC using the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, 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. In this case, AC is the hypotenuse, and AB and BC are the other two sides. Applying this theorem will allow us to calculate the length of AC, the final side of our triangle.

Once we know the lengths of all three sides of triangle ABC, we can calculate the trigonometric functions for the acute angles. The acute angles in triangle ABC are angle BAC and angle BCA. We'll use the trigonometric ratios sine, cosine, and tangent, which relate the angles of a right-angled triangle to the ratios of its sides. Understanding the relationship between the sides and angles in both the rectangle and the triangle is crucial for solving the problem accurately. By carefully analyzing the given information and applying the appropriate geometric and trigonometric principles, we can find the values of the trigonometric functions for the acute angles of triangle ABC. This initial understanding sets the stage for a detailed step-by-step solution.

Step 1: Determining the Length of Side AB

The first crucial step in solving this problem is to determine the length of side AB, which is one of the sides of the rectangle and also a side of the right-angled triangle ABC. To achieve this, we utilize the information provided about the perimeter of the rectangle. As we established earlier, the perimeter of a rectangle is the sum of the lengths of all its sides. In the case of rectangle ABCD, the perimeter is given as 10. We also know that the shorter side BC has a length of 2. Since opposite sides of a rectangle are equal, the side DA also has a length of 2.

Let's denote the length of side AB (and CD) as 'x'. The formula for the perimeter of a rectangle is 2 * (length + width). In our scenario, this translates to 2 * (AB + BC) = 10. Substituting the known values, we get 2 * (x + 2) = 10. This equation allows us to solve for x, which represents the length of side AB. Solving this equation involves simple algebraic manipulation. First, we can divide both sides of the equation by 2, which gives us x + 2 = 5. Next, we subtract 2 from both sides to isolate x, resulting in x = 5 - 2. Therefore, x = 3.

This calculation reveals that the length of side AB is 3. This is a significant piece of information because it provides us with the lengths of two sides of the right-angled triangle ABC: AB = 3 and BC = 2. With these lengths known, we can proceed to calculate the length of the third side, AC, which is the hypotenuse of the triangle. Determining the length of AB is a foundational step that allows us to apply the Pythagorean theorem and subsequently calculate the trigonometric functions of the acute angles in triangle ABC. This step demonstrates how understanding basic geometric properties, such as the perimeter of a rectangle, can lead to the solution of more complex problems involving triangles and trigonometry. With the length of AB now known, we are one step closer to finding the values of the trigonometric functions we seek.

Step 2: Calculating the Length of Hypotenuse AC using the Pythagorean Theorem

With the lengths of sides AB and BC now known, we can move on to the next crucial step: calculating the length of the hypotenuse AC of the right-angled triangle ABC. This is where the Pythagorean theorem comes into play, a fundamental concept in geometry that relates the sides of a right-angled triangle. As we discussed earlier, the Pythagorean theorem states that in a right-angled triangle, 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. In the context of triangle ABC, this translates to AC² = AB² + BC².

We have already determined that AB has a length of 3 and BC has a length of 2. Substituting these values into the Pythagorean theorem, we get AC² = 3² + 2². Squaring the values, we have AC² = 9 + 4. Adding these, we find AC² = 13. To find the length of AC, we need to take the square root of both sides of the equation. This gives us AC = √13. It's important to note that we only consider the positive square root since length cannot be negative.

Thus, we have successfully calculated the length of the hypotenuse AC to be √13. This is a crucial piece of information because it completes our knowledge of all three sides of triangle ABC: AB = 3, BC = 2, and AC = √13. With all side lengths known, we are now fully equipped to calculate the trigonometric functions of the acute angles in the triangle. The Pythagorean theorem is a powerful tool in geometry, and its application here demonstrates its practical utility in solving problems involving right-angled triangles. By using this theorem, we have bridged the gap between the known sides and the unknown hypotenuse, allowing us to proceed to the final step of calculating the trigonometric functions. The accurate calculation of AC is essential for the subsequent steps, as the trigonometric ratios are defined in terms of the ratios of the sides of the triangle.

Step 3: Calculating Trigonometric Functions for Acute Angles

Now that we know the lengths of all three sides of triangle ABC—AB = 3, BC = 2, and AC = √13—we can proceed to the final and most exciting part of the problem: calculating the values of the trigonometric functions for the acute angles. In a right-angled triangle, there are two acute angles, which are the angles less than 90 degrees. In triangle ABC, these angles are ∠BAC and ∠BCA. To calculate the trigonometric functions, we will use the basic definitions of sine, cosine, and tangent in relation to the sides of a right-angled triangle.

Let's begin by defining the trigonometric ratios. Sine (sin) of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Cosine (cos) of an angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. Tangent (tan) 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. These definitions are fundamental to trigonometry and provide the basis for calculating the trigonometric functions in our problem.

First, let's consider angle ∠BAC. For this angle, BC is the opposite side, AB is the adjacent side, and AC is the hypotenuse. Therefore:

sin(∠BAC) = Opposite / Hypotenuse = BC / AC = 2 / √13
cos(∠BAC) = Adjacent / Hypotenuse = AB / AC = 3 / √13
tan(∠BAC) = Opposite / Adjacent = BC / AB = 2 / 3

Next, let's consider angle ∠BCA. For this angle, AB is the opposite side, BC is the adjacent side, and AC is the hypotenuse. Therefore:

sin(∠BCA) = Opposite / Hypotenuse = AB / AC = 3 / √13
cos(∠BCA) = Adjacent / Hypotenuse = BC / AC = 2 / √13
tan(∠BCA) = Opposite / Adjacent = AB / BC = 3 / 2

It's often desirable to rationalize the denominators in trigonometric expressions. To do this, we multiply both the numerator and the denominator of the fractions with √13 by √13. This gives us:

sin(∠BAC) = (2 / √13) * (√13 / √13) = 2√13 / 13
cos(∠BAC) = (3 / √13) * (√13 / √13) = 3√13 / 13
sin(∠BCA) = (3 / √13) * (√13 / √13) = 3√13 / 13
cos(∠BCA) = (2 / √13) * (√13 / √13) = 2√13 / 13

We have now successfully calculated the values of the sine, cosine, and tangent functions for both acute angles in triangle ABC. This final step demonstrates the application of trigonometric ratios in solving geometric problems. By understanding the relationships between the sides and angles in a right-angled triangle and applying the appropriate trigonometric definitions, we can determine the values of these functions. This comprehensive approach, from understanding the problem to calculating the final trigonometric values, showcases the power of combining geometric and trigonometric principles in problem-solving.

Conclusion

In conclusion, we have successfully calculated the trigonometric functions for the acute angles of triangle ABC, given the properties of rectangle ABCD. This problem has taken us through a series of essential steps, from understanding the geometric properties of rectangles and triangles to applying the Pythagorean theorem and trigonometric ratios. By meticulously working through each step, we have demonstrated how fundamental mathematical principles can be combined to solve complex problems.

We began by understanding the problem statement and visualizing the geometric shapes involved. We then used the given perimeter and the length of the shorter side to determine the length of the longer side of the rectangle. This crucial step allowed us to find the lengths of two sides of the right-angled triangle ABC. Next, we applied the Pythagorean theorem to calculate the length of the hypotenuse, AC, completing our knowledge of all three sides of the triangle.

With the lengths of all sides known, we moved on to the final step: calculating the trigonometric functions for the acute angles. We used the definitions of sine, cosine, and tangent to find the ratios of the sides for each angle. We then rationalized the denominators to express the trigonometric values in their simplest forms. The results provide a comprehensive understanding of the trigonometric relationships within triangle ABC.

This exercise highlights the interconnectedness of different mathematical concepts. Geometry provides the foundation for understanding shapes and their properties, while trigonometry allows us to relate angles and sides in triangles. The Pythagorean theorem serves as a bridge between these concepts, enabling us to calculate unknown side lengths in right-angled triangles. By mastering these fundamental principles, one can approach a wide range of mathematical problems with confidence.

The ability to solve problems like this is not only valuable in academic settings but also has practical applications in various fields, including engineering, physics, and computer graphics. A thorough understanding of geometry and trigonometry enhances analytical and problem-solving skills, which are essential in many areas of life. This step-by-step guide serves as a valuable resource for students, educators, and anyone interested in deepening their understanding of mathematics. The combination of geometric insight and trigonometric calculations provides a powerful toolkit for tackling mathematical challenges. Through careful analysis and application of the appropriate principles, we can unravel complex problems and gain a deeper appreciation for the beauty and utility of mathematics.