What is the area of a triangle using sine?

The area of a triangle can be found using the formula: \( \frac{1}{2}ab \sin(C) \).

To understand this formula, let's break it down. In a triangle, \(a\) and \(b\) represent the lengths of two sides, and \(C\) is the angle between those two sides. The sine function (\(\sin\)) is a trigonometric function that relates the angle to the ratio of the opposite side to the hypotenuse in a right-angled triangle. However, it can also be used in non-right-angled triangles to find the area.

Imagine you have a triangle with sides \(a\) and \(b\), and the angle \(C\) between them. By using the formula \( \frac{1}{2}ab \sin(C) \), you are essentially calculating the area of a parallelogram formed by duplicating the triangle and then halving it, since a triangle is half of a parallelogram.

For example, if you have a triangle with sides \(a = 5\) cm, \(b = 7\) cm, and the angle \(C = 30^\circ\) between them, you can find the area as follows:
1. Calculate \(\sin(30^\circ)\), which is \(0.5\).
2. Plug the values into the formula: \( \frac{1}{2} \times 5 \times 7 \times 0.5 \).
3. This simplifies to \( \frac{1}{2} \times 35 \times 0.5 = 8.75 \) square centimetres.

This method is particularly useful when you don't have the height of the triangle but know two sides and the included angle. It’s a handy tool in your GCSE Maths toolkit!