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How does the law of sines help solve triangles?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — geometry This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

The law of sines is a fundamental principle in trigonometry used to solve triangles by relating the lengths of their sides to the sines of their opposite angles. The law is expressed as:123

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

where a, b, and c represent the side lengths and A, B, and C represent the corresponding opposite angles.12

To solve a triangle, you substitute known values into this equation and solve for the unknown quantities. The law of sines is specifically employed in the following scenarios:3

Two angles and one side: When you know two angles and the length of a side opposite one of them, you can calculate the length of the other opposite side.23
Two sides and one opposite angle: When you know the lengths of two sides and the angle opposite one of those sides, you can calculate the other opposite angle.23

For triangles on a sphere, a similar version of the law exists using the sines of the sides and angles.42