Science Fair Project Encyclopedia
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)]}](a/../upload/math/962754f928c55f5ea637d90b9459eb9a.png)
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{\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)]}](a/../upload/math/d687fde203051b42aad1add6383ec6ce.png)
- (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)]}](a/../upload/math/449ac3e7d08838acc3b2cfc658b7606f.png)
See also
10-26-2009 08:16:03
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details