Problem 37 - Precalculus

Question

Find the area of each triangle with measures given. $$b=100, c=150, \alpha=36^{\circ}$$

Step-by-Step Solution

Answer

The area of the triangle is 4408.5 square units.

Step 1: Understanding the Formula

To find the area of a triangle with sides and an included angle, we can use the formula: \( \text{Area} = \frac{1}{2}bc \sin(\alpha) \), where \(b\) and \(c\) are sides, and \(\alpha\) is the included angle.

Step 2: Substituting Given Values

Substitute the given values into the formula: \(b = 100\), \(c = 150\), and \(\alpha = 36^{\circ}\). Thus, the formula becomes: \( \text{Area} = \frac{1}{2} \times 100 \times 150 \times \sin(36^{\circ}) \).

Step 3: Calculating the Sine Value

Calculate \( \sin(36^{\circ}) \). Using a calculator, \( \sin(36^{\circ}) \approx 0.5878 \).

Step 4: Computing the Area

Substitute the sine value back into the area formula: \( \text{Area} = \frac{1}{2} \times 100 \times 150 \times 0.5878 = 4408.5 \).

Step 5: Conclusion

The area of the triangle is \( 4408.5 \) square units.

Key Concepts

Triangle Area

Finding the area of a triangle can be simple using the appropriate formula. When you know two sides and the included angle, you can find the area by utilizing the formula:

• \(\text{Area} = \frac{1}{2} bc \sin(\alpha)\)

Here, \(b\) and \(c\) are the lengths of the two sides, and \(\alpha\) is the included angle. This formula leverages trigonometry to account for the angle between the sides, giving a precise measure of the area.For example, given sides \(b = 100\) and \(c = 150\) with an angle \(\alpha = 36^{\circ}\), we can find the area as follows:

• First, calculate \(\sin(36^{\circ})\).
• Use the equation \(\text{Area} = \frac{1}{2} \times 100 \times 150 \times \sin(36^{\circ})\).
• Since \(\sin(36^{\circ}) \approx 0.5878\), the area calculates to approximately \(4408.5\) square units.

These steps highlight how trigonometry helps in determining areas beyond simple length measurements.

Sine Rule

The sine rule isn't just a pivotal tool for determining areas, but also for understanding relationships in triangles. It states:

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

This rule helps in finding unknown angles or lengths in any triangle, not only right-angled ones. However, in our triangle area context, the sine function aids in calculating the precise share of the enclosed angle:

• \(\sin(\alpha)\) is used in area calculations to adjust for the angle's effect on the triangle's size.
• It bridges the geometric dimensions of the shape with its angular properties.

In our previous example, knowing \(\alpha = 36^{\circ}\) lets us compute \(\sin(36^{\circ})\) and find the area effectively. This relationship offers insight into how angles influence measurements in geometry.

Geometry

Geometry, the study of shapes and their properties, provides us with essential tools like the sine rule and area formulas. In triangles, understanding the interplay of sides and angles is crucial:

• Triangles are foundational figures in geometry, and their properties lead to more complex shapes.
• When dealing with the area, it's important to note the type of triangle you're analyzing, such as scalene, isosceles, or right-angled.

For triangle area problems:

• Recognize the significance of the angle between known sides.
• Apply appropriate trigonometric functions, like sine, to find areas accurately.
• Use these principles to solve not just textual problems, but real-world scenarios involving land measurement, construction, etc.

Thus, by integrating trigonometry with geometric principles, you can solve many practical issues involving shapes and measurements.