Understanding the Sine Rule
The sine rule states that for any triangle with sides a, b, and c opposite to angles A, B, and C respectively, the relationship can be expressed as:
a/sin A = b/sin B = c/sin C.
This rule is particularly useful when solving triangles in the following scenarios:
- AAS (Angle-Angle-Side): When you know two angles and one side.
- ASA (Angle-Side-Angle): When you know one side and the two angles adjacent to it.
- SSA (Side-Side-Angle): When you know two sides and a non-included angle.
In these cases, you can use the sine rule to find unknown sides or angles. Remember that SSA can sometimes lead to ambiguous cases, so always check your results.
For example, in triangle ABC, if A = 30°, B = 45°, and a = 10 cm, we can find side b using the sine rule.
Key points to remember
- Sine rule: a/sin A = b/sin B = c/sin C.
- Apply when given AAS, ASA, or SSA.
- Useful for finding unknown sides or angles.
- Check for ambiguous cases in SSA.
- Always ensure angles are in the same triangle.
Worked example
Given triangle ABC with A = 30°, B = 45°, and a = 10 cm, find side b.
Using sine rule:
b/sin B = a/sin A
b = a * (sin B / sin A)
b = 10 * (sin 45° / sin 30°)
b = 10 * (√2/2 / 1/2)
b = 10 * √2
b ≈ 14.14 cm.