E-commerce
Calculating the Area of a Right Angled Triangle with Given Hypotenuse and Angle
Calculating the Area of a Right Angled Triangle with Given Hypotenuse and Angle
When dealing with triangles in mathematics or geometry, certain conditions can simplify the problem significantly. For example, when a triangle has a hypotenuse and one of its angles, it becomes a right-angled triangle, which allows for straightforward calculations, including the determination of the area of the triangle. In this article, we will explore how to calculate the area of a right-angled triangle given its hypotenuse and an angle of elevation.
Understanding the Triangle
A right-angled triangle is a triangle where one angle is exactly 90 degrees. The longest side, which is opposite the right angle, is called the hypotenuse. In this problem, we are given a hypotenuse of 55 units and an angle of elevation of 33 degrees, which is not an altitude but rather an angle from the hypotenuse.
Step-by-Step Calculation
Given the hypotenuse and an angle of elevation, we can use trigonometric functions to find the unknown sides of the triangle. Here's a detailed step-by-step process:
Step 1: Determine the Base of the Triangle
The base of the triangle can be found using the cosine function, which relates the adjacent side (base) to the hypotenuse:
Base hcosθ, where h is the hypotenuse and θ is the angle of elevation.
Given:
Hypotenuse (h) 55 units
Angle of elevation (θ) 33 degrees
Base (BC) 55cos(33°)
Using a calculator for precise value:
BC 55 * cos(33°) 55 * 0.83867 46.12685 ≈ 46.13 units
Step 2: Determine the Height of the Triangle
The height of the triangle can be found using the sine function, which relates the opposite side (height) to the hypotenuse:
Height hsinθ
Height (AB) 55 * sin(33°)
Using a calculator for precise value:
AB 55 * sin(33°) 55 * 0.54464 29.9552 ≈ 29.96 units
Calculating the Area of the Triangle
The area (A) of a right-angled triangle can be calculated using the formula:
A 1/2 * Base * Height
Substituting the values we found:
A 1/2 * 46.13 * 29.96
Calculating the area:
A 1/2 * 1381.4568 690.7284 ≈ 690.73 square units
Conclusion
This detailed process has shown us that with a hypotenuse of 55 units and an angle of elevation of 33 degrees, the area of the right-angled triangle can be calculated to be approximately 690.73 square units. Understanding the relationships between the sides and angles of a triangle using trigonometric functions is crucial for solving various geometric problems in mathematics and related fields.
Related Questions and Keywords
Keywords: right angled triangle, hypotenuse, angle of elevation, area calculation