Problem 41 - Precalculus with Limits

Question

In Exercises 39-44, find the area of the triangle having the indicated angle and sides. \(A\ =\ 43^{\circ}45'\), \(b\ =\ 57\), \(c\ =\ 85\)

Step-by-Step Solution

Answer

The area of the triangle will be obtained by calculating the expression from Step 4. Please use a scientific calculator, ensuring the calculator is set to degrees.

Step 1: Identify the known values

The values provided are: \(A = 43^\circ 45'\), \(b = 57\), and \(c = 85\). The aim is to find the area of the triangle, typically denoted by \(K\).

Step 2: Convert angle A to decimals

One degree is equivalent to 60 minutes. To convert angle A into decimals, divide the minutes by 60 and add the result to the degree value thus: \(A = 43 + \frac{45}{60} = 43.75 ^\circ\).

Step 3: Use the Cosine formula to find the area

Now, we can use the formula to find the area: \[ K = \frac{1}{2} * b * c * \sin A \]. Plugging in the known values we yield: \[ K = \frac{1}{2} * 57 * 85 * \sin 43.75 ^\circ \]

Step 4: Calculate the area

With all the values in place, proceed to complete the calculation to find the area of the triangle.

Key Concepts

Triangle Area Calculation

Understanding how to calculate the area of a triangle is a vital skill in trigonometry. There are various methods to determine the area based on the available information, especially in non-right triangles.
The specific formula used here is: \[ K = \frac{1}{2} \times b \times c \times \sin A \]
In this formula:

• \(K\) represents the area of the triangle.
• \(b\) and \(c\) are the lengths of two sides.
• \(\sin A\) is the sine of the angle that lies between them.

This formula is advantageous because it allows you to find the area with just two sides and the included angle.
For the given problem, with side lengths \(b = 57\) and \(c = 85\), and an included angle \(A = 43.75^{\circ}\) after conversion, the area calculation becomes a straightforward substitution into the formula. This method proves especially useful in non-right triangles where standard base-height calculations would be tricky.

Sine Rule

The Sine Rule is a powerful tool in trigonometry used primarily for solving triangles, especially those that are not right-angled. It relates the lengths of sides to the sines of their opposite angles. However, for this specific task, the Sine Rule directly helps in understanding the use of sine in area calculations.
To quickly recall, the Sine Rule states:

• \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)

Although the current problem doesn't directly require using the Sine Rule, knowing that sine relates so closely to side lengths and angles helps understand why \(\sin A\) is in the area formula. The sine of an angle effectively determines how much of the side contributes to the 'height' when conceptually dropping a perpendicular from one side to form a right triangle base, hence using \(\sin A\) in the area calculation. This relation underlines the importance of gaining comfort with trigonometric functions and their roles in geometric problems.

Degree Conversion

Before using trigonometric functions like the sine in calculations, it's paramount to have angles properly presented in decimal degrees. Many problems will give angles in degrees and minutes, requiring conversion.
For conversion, remember that:

• 1 degree = 60 minutes
• 1 minute = \(\frac{1}{60}\) of a degree

This means any minutes portion of an angle needs dividing by 60 to convert them to a decimal format.
In the given exercise, the angle is provided as \(43^{\circ} 45'\). Convert the 45 minutes by:

• \(\frac{45}{60} = 0.75\)

So the angle in decimal degrees becomes \(43.75^{\circ}\).
This conversion is crucial for two reasons: calculators and trigonometric tables generally require angles in decimal format, and having consistent units when performing mathematical operations simplifies solving the tasks accurately.