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

<math>\frac{a+b}{a-b} = \frac{\tan[\frac{1}{2}(A+B)]}{\tan[\frac{1}{2}(A-B)]}<math>

Derivation

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

<math>\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}}<math>

<math>\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)]}<math>

(See: Trigonometric identity)

<math>\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)]}<math>

<math>\frac{a+b}{a-b} = \frac{\tan[\frac{1}{2}(A+B)]}{\tan[\frac{1}{2}(A-B)]}<math>

See also

• Law of sines
• Law of cosines