law of tangents

In trigonometry, the law of tangents is a statement about arbitrary triangles in the plane. If two sides of a triangle are (lower-case) a and b and the angles opposite those sides are (capital) A and B, then the law of tangents states

\frac{a+b}{a-b} = \frac{\tan \frac{1}{2}(A+B)}{\tan \frac{1}{2}(A-B)}

Derivation

Start with (a + b)/(a - b). ((sin A)/a = (sin B)/b because of the law of sines):

\frac{a+b}{a-b} = \frac{a\cdot\frac{\sin A}{a} + b\cdot\frac{\sin B}{b}}{a\cdot\frac{\sin A}{a} - b\cdot\frac{\sin B}{b}}

\frac{a+b}{a-b} = \frac{\sin(A) + \sin(B)}{\sin(A) - \sin(B)} = \frac{2\sin \frac{1}{2}(A+B) \cdot \cos \frac{1}{2}(A-B)}{2\cos \frac{1}{2}(A+B) \cdot \sin \frac{1}{2}(A-B)}

(See: Trigonometric identity)

\frac{a+b}{a-b} = \frac{\sin \frac{1}{2}(A+B)}{\cos \frac{1}{2}(A+B)} \cdot \frac{\cos \frac{1}{2}(A-B)}{\sin \frac{1}{2}(A-B)}

\frac{a+b}{a-b} = \frac{\tan \frac{1}{2}(A+B)}{\tan \frac{1}{2}(A-B)}

Law Of Tangents

Derivation

See also