tangent of halved angle

The formulae


may be solved for  sinα  and  cosα, respectively.  One gets the equations


where the signs have to be chosen according to the quadrant where the angle α is.  Changing α to x2 and dividing these equations gives us the formula

tanx2=±1-cosx1+cosx. (1)

Also here one must chose the sign according to the quadrant of  x2.

We obtain two alternative forms of (1) when we multiply both the numerator and the denominator of the radicand the first time by  1-cosx  and the second time by  1+cosx; note that  1-cos2x=sin2x:

tanx2=1-cosxsinx, (2)
tanx2=sinx1+cosx (3)

Here,  sinx  determines the sign of the hand sides; it can be justified that it has always the same sign as tanx2.

Title tangent of halved angle
Canonical name TangentOfHalvedAngle
Date of creation 2013-03-22 17:00:32
Last modified on 2013-03-22 17:00:32
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Derivation
Classification msc 26A09
Related topic DerivationOfHalfAngleFormulaeForTangent
Related topic GoniometricFormulae