law of tangents

Fig. 1 - A triangle.

In trigonometry, the law of tangents is a statement about arbitrary triangles in the plane.

In Figure 1, a, b, and c are the lengths of the three sides of the triangle, and α, β, and γ are the angles opposite those three respective sides. The law of tangents states that

The law of tangents, although not as commonly known as the law of sines or the law of cosines, is just as useful, and can be used in any case where you know either two sides and an angle, or two angles and a side.

Proof

To prove the law of tangents we can start with the law of sines:

Writing q for this common value, we get Failed to parse (unknown function\scriptstyle): \scriptstyle{a\,=\,q\sin\alpha} , Failed to parse (unknown function\scriptstyle): \scriptstyle{b\,=\,q\sin\beta} , so

Using the trigonometric identity

for Failed to parse (unknown function\scriptstyle): \scriptstyle{x\,=\,\alpha}

and Failed to parse (unknown function\scriptstyle): \scriptstyle{y\,=\,\pm\beta}
we get 

See also

• Law of sines
• Law of cosines

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