Problem 14 - Algebra 2 and Trigonometry

Question

In \(9-14,\) find the area of each triangle to the nearest tenth. In \(\triangle D E F, e=336, f=257, \mathrm{m} \angle D=122\)

Step-by-Step Solution

Answer

The area of \( \triangle DEF \) is approximately 35782.3 square units.

Step 1: Understand the Problem

We are given a triangle \( \triangle DEF \) with side \( e = 336 \), side \( f = 257 \), and angle \( \angle D = 122^{\circ} \). We need to find the area of this triangle.

Step 2: Choose the Formula for the Area

We will use the formula for the area of a triangle given two sides and the included angle: \( \text{Area} = \frac{1}{2} \, e \, f \, \sin(D) \).

Step 3: Convert Angle to Radians

Since many calculators require angles in radians to calculate the sine, convert \( 122^{\circ} \) to radians: \( 122^{\circ} = \frac{122 \pi}{180} \) radians.

Step 4: Calculate the Sine of the Angle

Use a calculator to find \( \sin(122^{\circ}) \). First, ensure your calculator is in degree mode or has converted 122 degrees to radians. \( \sin(122^{\circ}) \approx 0.829 \).

Step 5: Apply the Area Formula

Substitute the values into the area formula: \[ \text{Area} = \frac{1}{2} \times 336 \times 257 \times 0.829 \approx 35782.26 \].

Step 6: Round the Area to the Nearest Tenth

Round 35782.26 to the nearest tenth, which gives \( 35782.3 \).

Key Concepts

Triangle Area Calculation

Understanding how to find the area of a triangle using its sides and an angle is a key skill in trigonometry. There are various formulas offered, but when we have two sides and the included angle, the most effective one is:

• Area = \( \frac{1}{2} \times e \times f \times \sin(D) \)

This formula requires knowing the lengths of two sides, \( e \) and \( f \), and the measure of the angle between them, \( \angle D \). The multiplication of these values gives us the triangle's area in units squared. A critical aspect to remember is that the sine function hinges on the angle, so its accurate calculation is vital. Ensuring the calculator setting correctly interprets the angle is essential, whether degrees or radians.

Sine Function

The sine function is one of the fundamental trigonometric functions, widely used in triangle calculations such as finding areas. This function takes an angle as input and gives a number as output, which represents a ratio of the lengths of the sides of a right triangle.

In the context of triangle area calculation, knowing the sine of the angle between two sides allows us to determine how those sides contribute to the total area. In simpler terms, sine helps adjust the two side lengths for the angular displacement, ensuring an accurate measurement of the area.

• The range of the sine function is from -1 to 1, meaning the output will always stay within these bounds.
• Sine works with angles measured in degrees or radians.
• The function is periodic, repeating its values in predictable cycles typically every 360 degrees or \( 2\pi \) radians.

Angle Conversion

Conversion between degrees and radians is often needed in trigonometry because mathematical calculations involving trigonometric functions usually require angles to be in radians.

To convert degrees to radians, the formula used is:

• Radians = \( \frac{\text{Degrees} \times \pi}{180} \)

In our example, converting the angle \(122^{\circ}\) into radians makes it suitable for most trigonometric calculations, especially when using a calculator set to radian mode.

Knowing how to perform this conversion helps in ensuring calculations are accurate, as it harmonizes the angle measurements with the format expected by many mathematical formulas. Remember, the conversion process links both units to equivalent circular measurements, offering an entirely different yet equally useful way to comprehend and apply angles.