Solving Triangles Using the Law of Sines
Rule

Law of Sines

For any triangle, the ratio of the sine of an angle to its opposite side is constant.
Triangle with one interactive vertex
Based on the characteristics of the diagram, the following relation can be proven true.

An alternative way to write the Law of Sines is involving the ratio of a side length to the sine of its opposite angle.

Proof

Acute Triangles

Consider an acute triangle with height drawn from one of its vertices. Because is perpendicular to the base, the original triangle is split into two right triangles.

Triangle split into two right triangles
In these two right triangles, and are both opposite angles to Therefore, by applying the definition of the sine ratio to and it is possible to relate the sine of these angles with the side lengths and
Showing sine of angles A and B
Next, can be isolated and written in terms of the corresponding side length and angle, for both right triangles.
By the Transitive Property of Equality, it can be stated that and are equal.
Finally, the obtained equation can be rearranged to obtain the desired result.
Simplify
By following the same procedure but drawing the height from vertex it can be shown that Putting these two results together, the Law of Sines is obtained.

Obtuse Triangles

An obtuse triangle will now be considered.

Obtuse Triangle

This proof is very similar to the proof for acute triangles, but it uses an interior and an exterior height. First, the height from the vertex where the obtuse angle is located will be drawn. Just as before, this generates two right triangles.

Drawing inner altitude for an obtuse triangle
The sine ratio will be written for these right triangles.
Showing sine of angles A and C
In the equations, can be isolated.
By the Transitive Property of Equality, it can be stated that and are equal.
The obtained equation can be rearranged to obtain the desired result.
Simplify
Next, the exterior height from vertex will be drawn. Let be the point of intersection of this height and the extension of
Obtuse Triangle Exterior Height
Here, and form a linear pair and are therefore supplementary angles. Because the sine of supplementary angles is the same, the sine of equals the sine of Also, using the sine ratio on it can be stated that the sine of is the ratio of to
By the Transitive Property of Equality, the sine of can be written in terms of and
Now will be considered. By using the sine ratio, it follows that the sine of is the ratio of to
Obtuse Triangle
Now, can be written in terms of and and in terms of and
The Transitive Property of Equality can be used one more time.
If only is considered, then can be named as
Obtuse Triangle
This allows to be written as Therefore, and this equation can be rearranged to obtain the desired formula.
Simplify
Finally, this result can be combined with the previous determination to derive the Law of Sines.

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