Problem 74 - A Graphical Approach to Precalculus with Limits

Question

Find the area of each triangle. \(A=42.5^{\circ}, b=13.6\) meters, \(c=10.1\) meters

Step-by-Step Solution

Answer

The area of the triangle is approximately 45.95 square meters.

Step 1: Identify the formula for the area of a triangle

To find the area of a triangle when two sides and an included angle are given, we use the formula: \[ \text{Area} = \frac{1}{2}bc \sin A \] where \( b \) and \( c \) are the lengths of the sides, and \( A \) is the measure of the included angle.

Step 2: Convert angle into radians

Since the angle provided is in degrees, we need to convert it into radians for calculation purposes. Use the formula: \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \] Thus, \[ 42.5^{\circ} = 42.5 \times \frac{\pi}{180} \approx 0.7418 \text{ radians.} \]

Step 3: Calculate the sine of the angle

Now, compute the sine of the angle \( A \) in radians. \[ \sin(0.7418) \approx 0.6704 \]

Step 4: Substitute values into the formula

Substitute the known values into the area formula:\[ \text{Area} = \frac{1}{2} \times 13.6 \times 10.1 \times 0.6704 \]

Step 5: Simplify the expression to find the area

Calculate the area using the values substituted: \[ \text{Area} \approx \frac{1}{2} \times 13.6 \times 10.1 \times 0.6704 \] \[ \text{Area} \approx 45.95 \text{ square meters}\]

Key Concepts

Area of a Triangle

To understand how to find the area of a triangle when given two sides and the included angle, you start with the specific formula used for such cases. This formula is essential because it utilizes the properties of trigonometry to deliver accurate results. The formula is:

• \[ \text{Area} = \frac{1}{2}bc \sin A \]

In this formula, the variables \( b \) and \( c \) represent the lengths of the two sides, and \( A \) is the angle between those two sides. This method is particularly useful when the angle is known, as it allows you to involve trigonometric functions for precise determination. Understanding this approach is important because it applies differently than the simpler base-height formula used in right-angled triangles.
The reason the sine function is used in this formula is due to its ability to relate angles and sides in a non-right-angled triangle, thus offering a wider scope of application in real-world scenarios.

Radian Conversion

Understanding how to convert degrees to radians is an important skill in trigonometry. This conversion is crucial because most functions in advanced mathematics and sciences use radian measures. To convert an angle given in degrees to radians, you apply the following conversion formula:

• \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]

This formula arises from the fact that a full circle measures \( 360^{\circ} \) in degrees, which equals \( 2\pi \) radians. Therefore, 1 degree equals \( \frac{\pi}{180} \) radians. To exemplify, an angle of \( 42.5^{\circ} \) converts to approximately \( 0.7418 \) radians using this method. Knowing how to switch between these units helps in solving problems across various mathematical topics that prefer or require radian measure as the standard.

Sine Function

The sine function is a fundamental component in trigonometry, representing the ratio of the length of the side of a right-angled triangle opposite a given angle to the hypotenuse. In mathematical terms, if \( \theta \) is an angle of a right triangle:

• \[ \sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \]

However, in the context of non-right triangles or for angles given in radians, the sine function can be evaluated using a calculator or trigonometry tables since it provides values based on circular (unit circle) interpretations. Take for example, the angle converted to radians in the earlier section, \( 0.7418 \), has a sine value of approximately \( 0.6704 \).
The sine function's use in the area formula \( \text{Area} = \frac{1}{2}bc \sin A \) is based on its ability to handle non-right triangles by adequately mapping relationships between angles and their corresponding sides in a circle.