How do you find a side in a non-right-angled triangle using the sine rule?

To find a side in a non-right-angled triangle using the sine rule, use the formula: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).

The sine rule is a powerful tool in trigonometry for solving problems involving non-right-angled triangles. It relates the lengths of the sides of a triangle to the sines of its angles. To use the sine rule to find a missing side, you need to know at least one angle-side pair and one additional angle or side.

For example, suppose you have a triangle with sides \(a\), \(b\), and \(c\), and angles \(A\), \(B\), and \(C\) opposite these sides, respectively. If you know the lengths of sides \(a\) and \(b\) and the measure of angle \(A\), you can find side \(c\) if you also know angle \(C\). The sine rule states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides and angles in the triangle.

To find side \(c\), rearrange the sine rule formula to solve for \(c\):
\[ c = \frac{a \cdot \sin C}{\sin A} \]

Make sure your angles are in degrees (or radians, depending on your calculator settings) and use the sine function to find the sine values. Plug in the known values and solve for the unknown side. This method allows you to find the missing side accurately, provided you have the necessary information about the other sides and angles.