How do you use the sine rule to solve for an angle?

To use the sine rule to solve for an angle, rearrange the formula and use the given sides and angles.

The sine rule states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. This can be written as:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

where \(a\), \(b\), and \(c\) are the lengths of the sides, and \(A\), \(B\), and \(C\) are the angles opposite those sides, respectively.

To solve for an angle, you need to know at least one side length and its opposite angle, as well as another side length. For example, if you know sides \(a\) and \(b\) and angle \(A\), you can find angle \(B\) using the rearranged sine rule:

\[ \sin B = \frac{b \cdot \sin A}{a} \]

Once you have this equation, you can solve for \(\sin B\) and then use the inverse sine function (also known as arcsine) to find the measure of angle \(B\):

\[ B = \sin^{-1} \left( \frac{b \cdot \sin A}{a} \right) \]

Make sure your calculator is set to the correct mode (degrees or radians) based on the context of your problem. This method allows you to find unknown angles in non-right-angled triangles, which is particularly useful in various geometric and trigonometric problems.