The sine Rule
Introduction
The Sine Rule is a powerful tool used to solve non-right-angled triangles. It allows us to find unknown angles or sides of a triangle when we have enough information. The rule relates the ratios of the lengths of the sides of a triangle to the sines of their opposite angles.
The Sine Rule is used in two main cases:
1. To find a missing side when we know two angles and one side (AAS or ASA).
2. To find a missing angle when we know two sides and one non-included angle (SSA).
The Sine Rule Formula
For any triangle \(ABC\), with sides \(a\), \(b\), and \(c\) opposite angles \(A\), \(B\), and \(C\) respectively, the Sine Rule is given by:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
When to use the sine Rule
Case 1: Angle-Angle-Side (AAS) or Angle-Side-Angle (ASA)
- You are given two angles and one side.
- Use the Sine Rule to find missing sides or angles.
Case 2: Side-Side-Angle (SSA)
- You are given two sides and an angle that is not between the two sides.
- Use the Sine Rule to find missing angles or sides, but be cautious as this can sometimes lead to the ambiguous case.
Interactive Graph
Example 1: Finding Side length
In triangle \(ABC\), we know that: Angle \(A = 40^\circ\), Angle \(B = 70^\circ\), Side \(a = 8 \, \text{cm}\). Find the length of side \(b\).
Example 2: Finding Angle
In triangle \(DEF\), we know that: Side \(d = 10 \, \text{cm}\), Side \(e = 12 \, \text{cm}\), Angle \(D = 35^\circ\). Find angle \(E\).
Ambiguous Case in the Sine Rule
In certain cases where you are given two sides and a non-included angle (SSA), there may be:
- No solution if the conditions of the triangle are impossible.
- One solution if the triangle is well-defined.
- Two solutions (ambiguous case) when two different triangles satisfy the given conditions.
This ambiguity arises because the sine function gives the same value for two different angles in the range \(0^\circ\) to \(180^\circ\). When solving for an angle, always check if a second solution is possible by considering:
\[
180^\circ – \text{first solution}
\]
Example 3 : Ambiguous Case
In triangle \(GHI\), we know that: Side \(g = 9 \, \text{cm}\), Side \(h = 10 \, \text{cm}\), Angle \(G = 30^\circ\). Find angle \(H\).
Practice Questions
2. In triangle \(PQR\), side \(p = 8 \, \text{cm}\), side \(q = 6 \, \text{cm}\), and angle \(P = 50^\circ\). Find side \(r\).
3. In triangle \(ABC\), side \(a = 12 \, \text{cm}\), angle \(A = 30^\circ\), and angle \(B = 45^\circ\). Find side \(b\).
Conclusion