Calculating Length P In A House Diagram Using Trigonometry
In the realm of mathematics, particularly in trigonometry and geometry, practical problems often require us to apply theoretical concepts to real-world scenarios. One such scenario is the calculation of unknown dimensions in architectural designs or structural diagrams. This article delves into a specific problem involving a house diagram with a triangular roof, where we need to determine the length 'p' using trigonometric principles. We will explore the given dimensions, the angles involved, and the step-by-step methodology to arrive at the solution, ensuring a comprehensive understanding of the underlying concepts and their application.
Problem Statement
We are presented with a diagram of a house featuring a triangular roof. The base of the triangular roof measures 1.95 meters. Two angles within the triangle are given: 53° and 38°. Our task is to calculate the length 'p,' which represents one of the sides of the triangle. The final answer must be expressed in meters, rounded to two decimal places. This problem necessitates the use of trigonometric relationships within triangles, specifically the sine rule, to establish the relationship between the sides and angles and, subsequently, solve for the unknown length 'p'. Understanding the sine rule and its applicability is crucial for tackling this problem effectively.
Understanding the Sine Rule
The sine rule is a fundamental trigonometric principle that establishes a relationship between the sides of a triangle and the sines of their opposite angles. In any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Mathematically, this can be expressed as:
a / sin(A) = b / sin(B) = c / sin(C)
where a, b, and c are the lengths of the sides of the triangle, and A, B, and C are the angles opposite those sides, respectively. The sine rule is particularly useful when we have information about two angles and one side, or two sides and one angle, and we need to find the remaining sides or angles. In our problem, we are given two angles (53° and 38°) and one side (the base of 1.95 meters). To apply the sine rule effectively, we need to identify the angle opposite the side we are trying to find ('p') and the angle opposite the known side (1.95 meters). This will allow us to set up the ratios and solve for 'p'.
Step-by-Step Solution
1. Calculate the Third Angle
In any triangle, the sum of the interior angles is always 180°. We are given two angles, 53° and 38°. Therefore, we can calculate the third angle (let's call it angle C) as follows:
C = 180° - 53° - 38° = 89°
This third angle, 89°, is crucial because it is opposite the known side, the base of the triangle (1.95 meters). Now we have a complete set of angles for the triangle, which is essential for applying the sine rule.
2. Identify the Angle Opposite the Unknown Side
We need to find the length 'p.' Looking at the diagram, we can identify the angle opposite side 'p' as 38°. This angle is one of the given angles in the problem. Knowing the angle opposite 'p' is crucial for setting up the sine rule equation correctly.
3. Apply the Sine Rule
Now that we have the angle opposite the unknown side 'p' (38°) and the angle opposite the known side (89°), we can apply the sine rule. Let's denote the side opposite the 89° angle as 'a' (which is 1.95 meters) and the side 'p' as 'b.' The sine rule equation becomes:
p / sin(38°) = 1.95 / sin(89°)
This equation establishes the proportional relationship between the sides and the sines of their opposite angles. The next step is to isolate 'p' and solve for its value.
4. Solve for 'p'
To solve for 'p,' we need to isolate it on one side of the equation. We can do this by multiplying both sides of the equation by sin(38°):
p = (1.95 * sin(38°)) / sin(89°)
Now, we can use a calculator to find the values of sin(38°) and sin(89°). sin(38°) is approximately 0.6157, and sin(89°) is approximately 0.9998. Plugging these values into the equation, we get:
p = (1.95 * 0.6157) / 0.9998
5. Calculate the Value of 'p'
Performing the multiplication and division, we get:
p ≈ 1.2012 / 0.9998
p ≈ 1.20 meters
Therefore, the length 'p' is approximately 1.20 meters.
6. Round to Two Decimal Places
The problem requires the answer to be rounded to two decimal places. Since our calculated value is already very close to 1.20, rounding to two decimal places gives us:
p ≈ 1.20 meters
This is our final answer, expressed to the required precision.
Conclusion
In conclusion, by applying the principles of trigonometry, specifically the sine rule, we have successfully calculated the length 'p' in the house diagram. The steps involved calculating the third angle, identifying the angle opposite the unknown side, applying the sine rule, solving the equation, and rounding the final answer to two decimal places, demonstrated a comprehensive understanding of the problem-solving process. This problem highlights the practical application of trigonometric concepts in real-world scenarios, such as architectural design and structural calculations. Mastering these skills is crucial for students and professionals in fields like mathematics, engineering, and architecture. Understanding trigonometric rules allows for accurate determination of unknown lengths and angles, essential for precision and safety in various applications. This exercise underscores the importance of mathematical principles in solving practical problems and reinforces the significance of trigonometry as a powerful tool in geometry and related fields. By systematically applying the sine rule and performing the necessary calculations, we have arrived at the solution, showcasing the effectiveness of trigonometric methods in determining unknown dimensions. The length 'p' is therefore approximately 1.20 meters, accurately calculated and rounded to the specified decimal places. This result not only answers the specific problem but also demonstrates a broader understanding of how trigonometry can be used to solve complex geometrical challenges. Through this detailed explanation and step-by-step solution, we aim to provide a clear and accessible guide for anyone facing similar problems, reinforcing the value of mathematical knowledge and its real-world applications.
