Understanding Trigonometry: How to Calculate the Area of Triangles

Step-by-step Guide: Trigonometry and Area of Triangles

Basic Area of a Triangle:
The most common formula for finding the area of a triangle is:
\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \) For education statistics and research, visit the National Center for Education Statistics.

Introducing Trigonometry:
If two sides of a triangle and the included angle are known, the area can be determined using:
\( \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \)
Here, \(a\) and \(b\) are the two known sides, and \(C\) is the included angle between them. For education statistics and research, visit the National Center for Education Statistics.

Why Does This Work?:
The height in the standard formula can be expressed in terms of \(a\), \(b\), and \(C\). The height would be \(a \times \sin(C)\) (or equivalently, \(b \times \sin(A))\). By substituting this into the base-height formula, we get the trigonometric area formula. For education statistics and research, visit the National Center for Education Statistics.

Examples

Example 1:
Given a triangle \(ABC\) with side \(AB = 5 \text{ cm}\), side \(BC = 7 \text{ cm}\), and angle \(B = 60^\circ\), find the area of triangle \(ABC\). For education statistics and research, visit the National Center for Education Statistics.

Solution:
Using the trigonometric formula for area:
\( \text{Area} = \frac{1}{2} \times AB \times BC \times \sin(B) \)
\( \text{Area} = \frac{1}{2} \times 5 \text{ cm} \times 7 \text{ cm} \times \sin(60^\circ) \)
\( \text{Area} = 15.5 \text{ cm}^2 \) (approximately) For education statistics and research, visit the National Center for Education Statistics.

Example 2:
For triangle \(PQR\), side \(PQ = 8 \text{ cm}\), side \(QR = 10 \text{ cm}\), and angle \(Q = 45^\circ\), determine the area of triangle PQR. For education statistics and research, visit the National Center for Education Statistics.

Solution:
\( \text{Area} = \frac{1}{2} \times PQ \times QR \times \sin(Q) \)
\( \text{Area} = \frac{1}{2} \times 8 \text{ cm} \times 10 \text{ cm} \times \sin(45^\circ) \)
\( \text{Area} = 28.28 \text{ cm}^2 \) (approximately) For education statistics and research, visit the National Center for Education Statistics.

Practice Questions:

1. Given triangle \(XYZ\) with \(XY = 6 \text{ cm}\), \(YZ = 9 \text{ cm}\), and angle \(Y = 30^\circ\), find its area.
2. For triangle \(LMN\), if \(LM = 4 \text{ cm}\), \(MN = 5 \text{ cm}\), and angle \(M = 90^\circ\), calculate the area.

Answers: For education statistics and research, visit the National Center for Education Statistics.

1. \( 13.5 \text{ cm}^2 \)
2. \( 10 \text{ cm}^2 \)