Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the area of the triangle having the indicated angle and sides. \(C=120^{\circ}, \quad a=4, \quad b=6\)

Step-by-Step Solution

Answer

\(\frac{1}{2} \times 4 \times 6 \times \sin(\frac{2\pi}{3}) = 6 \times \sqrt{3}\)

Step 1: Convert Angles to Radians

The value of the angle C is given in degrees. Trigonometric calculations, however, typically require angles to be in radians. To convert from degrees to radians, the following formula is used: radians=\(\frac{degrees * \pi}{180}\). So in this case, the conversion would be done as \(C_{radians} = \frac{120 * \pi}{180} = \frac{2\pi}{3}\).

Step 2: Apply the Sine Rule

The formula for the area of a triangle using the sine rule is: Area = \(\frac{1}{2} \times a \times b \times \sin(C)\). So, the area of the triangle would be calculated as: \(Area = \frac{1}{2} * 4 * 6 * \sin(\frac{2\pi}{3})\).

Step 3: Perform the Calculation

Next, perform the calculation from step 2. This gives: \(Area = 6 * \sin(\frac{2\pi}{3})\). Calculate the sine of \(\frac{2\pi}{3}\), then multiply by 6 to find the area of the triangle.

Key Concepts

Area of Triangle

The area of a triangle can be calculated in different ways depending on the information available. In our example, we are using two sides and the included angle to find the area. This method is useful because it allows a calculation when the height of the triangle is not known. The formula used is:

• Area = \( \frac{1}{2} \times a \times b \times \sin(C) \)

Here, \(a\) and \(b\) are the lengths of the two sides, and \(C\) is the angle between them. By calculating \(\sin(C)\), you can find the proportion of the two sides that contribute to the height of the triangle. This method simplifies finding the area without needing a right angle.

Degree to Radian Conversion

Angles in trigonometry are more often than not expressed in radians rather than degrees. This is because radians provide a natural way to describe angles in terms of the circle's properties. To convert an angle given in degrees to radians, use the conversion formula:

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

In the problem, the angle \(C\) is \(120^{\circ}\). Using the formula:

• \(C_{radians} = \frac{120 \times \pi}{180} = \frac{2\pi}{3}\)

Remembering this conversion will help in settings where trigonometric functions are calculated, as functions like sine and cosine typically utilize radian measures.

Sine Rule

The sine rule in trigonometry is not only used for finding sides or angles in non-right angled triangles, but also for calculating the area, as in the solution. The rule states:

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

For the area, however, we use a different variation:

• Area = \( \frac{1}{2} \times a \times b \times \sin(C) \)

This form of the sine rule is efficient because it allows us to compute an area directly from simple measurements of two sides and an angle included between them. Knowing the angle \(C\) in radians, one can substitute into the formula and perform the multiplication and sine calculation to get the area easily.