Problem 37 - Algebra and Trigonometry

Question

In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit. $$ C=124^{\circ}, a=4 \text { meters, } b=6 \text { meters } $$

Step-by-Step Solution

Answer

The area of the triangle, rounded to the nearest square unit, is 12 square meters.

Step 1: Convert Degrees to Radians

Given angle C in degrees is 124. Convert this to radians because the argument for the sin function is in radians when it's used in calculations. Use the relation \( pi \) rad = 180 degrees to convert: \( C = 124 * (pi / 180) \) rad

Step 2: Use the Given Measurements

Plug in the given measurements for a, b, and angle C (now in radians) into the formula for the area of a triangle: \( area = 1/2 * a * b * sin(C) = 1/2 * 4 meters * 6 meters * sin(124*pi/180) \)

Step 3: Compute and Round Off

Compute the product and round the answer to the nearest square unit.

Key Concepts

Degrees to Radians Conversion

When dealing with angles, especially in trigonometry or geometry involving calculations, it's often useful to convert degrees into radians. This is because most trigonometric functions, like the sine function, expect angles to be in radians when performing calculations. To convert degrees to radians, you use the conversion ratio where \( \pi \) radians is equivalent to 180 degrees. Here’s the simple formula to make the conversion:

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

For instance, if you have an angle of 124 degrees, like in our triangle problem, you would convert it to radians as follows:

\[ 124 \times \frac{\pi}{180} \]

This conversion ensures that calculations using trigonometric functions are accurate, as the sine calculator will interpret the angle correctly.

Sine Function in Trigonometry

The sine function is one of the foundational elements in trigonometry. It relates to the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. However, when dealing with any triangle, the sine function helps in various calculations such as finding the area. In our triangle example, we use the formula for the area of a triangle given two sides and the included angle:

\[ \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \]

Here, \( a \) and \( b \) are the side lengths, while \( C \) is the angle between them in radians.

• The sine function returns a value between -1 and 1.
• It helps in scaling the size of the triangle accurately when multiplied with the side lengths.
• The output of the sine function represents how much of the maximum possible area the triangle takes up given its side lengths and included angle.

This is why converting angle measurements to radians is crucial, as it directly affects the calculation of sine and hence the area.

Triangle Side Lengths Measurement

Understanding the components forming a triangle is essential in solving problems related to area, especially in exercises like the one given where side lengths of the triangle are used. In our example, the triangle's sides \( a \) and \( b \) are 4 meters and 6 meters, respectively. These measurements are crucial for computing the area using the formula mentioned earlier.

The formula, which relies on the side lengths and the sin of the included angle, is effective because:

• It doesn't require the third side or the height of the triangle, making it efficient for such measurements.
• Knowing two sides and the included angle is sufficient to compute the area accurately.

By measuring side lengths accurately, any error in the computed area will be minimized. For practical purposes, using consistent units is important, like meters in our exercise, to ensure all parts of the formula are compatible.

Achieving precision in measuring these sides is important for students to understand real-world applications of trigonometry and geometric calculations.