Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the area of the triangle having the indicated angle and sides. \(B=75^{\circ} 15^{\prime}, \quad a=103, \quad c=58\)

Step-by-Step Solution

Answer

To find the final area, calculate \( \frac{1}{2}*103*58*Sin(75.25^{\circ})\). The area value will be the required solution.

Step 1: Convert Angle to Degrees

B is provided as degrees and minutes. We want to convert it all into degrees for easier calculation. The conversion is done using the formula: Degree + (Minutes/60). Hence, \( B = 75 + \frac{15}{60} = 75.25 ^{\circ}\).

Step 2: Calculate Area

Now we will substitute the values of a, c, and B into the formula. Hence, \( Area = \frac{1}{2}*103*58*Sin(75.25^{\circ})\).

Step 3: Final Calculation

Next, plug all values to get the final area. Since the Sin of an angle is unitless, the area will be in square units of the side lengths.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions play a crucial role in geometry, especially when dealing with triangles. The primary trigonometric function utilized in this exercise is the sine function. Sine, along with cosine and tangent, helps to relate angles to side lengths in triangles. Particularly, the sine function is useful in calculating the area of a triangle when two sides and the included angle are known. This is known as the principle of using half of the sine of the angle. The formula used in this exercise is:

• Area = \( \frac{1}{2} ab \sin(C) \)

Where \(a\) and \(b\) are the lengths of the sides, and \(C\) is the included angle between those sides. This function gives us a way to determine an area without needing a right-angle triangle, which is quite useful in many geometric calculations.

Exploring Angle Conversion

Angle conversion is essential for performing accurate geometric calculations. Angles can be expressed in various units, most commonly in degrees or in degrees and minutes. Converting them into a single unit is necessary for simplifying mathematical computations. In this exercise, we converted the angle from degrees and minutes into just degrees using the formula:

• Degrees = Degrees + (Minutes/60)

This formula allows us to include partial degrees, making calculations much simpler. For example, converting \(75^{\circ} 15^{\prime}\) to degrees involves adding \(75\) degrees to \(15/60\) degrees, resulting in \(75.25^{\circ}\). This unified format helps streamline the process of applying trigonometric functions effectively.

Basics of Geometry Calculations

Geometry calculations often involve determining the area, perimeter, or volume of shapes. Here, we focus on finding the area of a triangle using available side lengths and angles. The method employed, involving trigonometric functions, circumvents the need for perpendicular height measurements, which aren't always accessible in non-right triangles. In this particular exercise, after converting the angle to degrees and using the trigonometric area formula:

• Area = \( \frac{1}{2} \cdot 103 \cdot 58 \cdot \sin(75.25^{\circ}) \)

we end up with the final area measurement in square units. These calculations show how leveraging trigonometry can provide solutions to complex geometrical problems with ease and precision, especially when right angles don't apply. Understanding these calculation methods is foundational for tackling more advanced geometric challenges.