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How can the law of sines solve a triangle that is not a right triangle?

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Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides. — trigonometry Based on the definitions, various simple relationships exist among the functions. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A/cos A.

The law of sines is a fundamental principle in trigonometry that applies to any triangle, not just right triangles. It states that the lengths of the sides are proportional to the sines of their opposite angles, expressed as:

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

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

You can use this law to solve for unknown parts of a triangle in two primary scenarios:

Finding an angle: When you know the lengths of two sides and the measure of an angle opposite one of those sides, you can calculate the other opposite angle.2
Finding a side: When you know two angles and the length of one side opposite one of those angles, you can calculate the length of the other side opposite the second known angle.2

This relationship ensures that the ratio of each side to the sine of its opposite angle remains constant for all three sides and angles of the triangle.3