derivation of half-angle formulae for tangent


Start with the angle duplication formula

tan(x)=2tan(x/2)1-tan2(x/2).

Cross-multiply and move terms around:

tan(x)tan2(x/2)+2tan(x/2)=tan(x)

Divide by tan(x):

tan2(x/2)+2tan(x/2)tanx=1

Add 1/tan2(x) to both sides:

tan2(x/2)+2tan(x/2)tanx+1tan2(x)=1+1tan2(x)

CompletePlanetmathPlanetmathPlanetmath the square (http://planetmath.org/CompletingTheSquare):

(tan(x/2)+1tan(x))2=1+1tan2(x)

Take a square root and move a term to obtain the half-angle formula:

tan(x/2)=1+1tan2(x)-1tan(x)

To derive the other forms of the formula, we start by substituting sin(x)/cos(x) for tan(x):

tan(x/2)=1+cos2(x)sin2(x)-cos(x)sin(x)

Put the stuff inside the square root over a common denominator:

sin2(x)+cos2(x)sin2(x)-cos(x)sin(x)

Recall that sin2(x)+cos2(x)=1. Hence, we may get rid of the square root:

1sinx-cos(x)sin(x)

Putting the terms over a common denominator, we obtain our formula:

tan(x/2)=1-cos(x)sin(x)

To obtain the next formula, multiply both numerator and denominator by 1+cos(x):

(1-cos(x))(1+cos(x))sin(x)(1+cos(x))

Multiply out the numerator and simplify:

1-cos2(x)sin(x)(1+cos(x))

Note that the numerator equals sin2(x):

sin2(x)sin(x)(1+cos(x))

Cancel a common factor of sin(x) to obtain the formula

tan(x/2)=sin(x)1+cos(x).

To obtain the last formula, multiply the previous two formulae:

tan2(x/2)=1-cos(x)sin(x)sin(x)1+cos(x)

Cancel the common factor of sin(x):

tan2(x/2)=1-cos(x)1+cos(x)

Take the square root of both sides to obtain the formula

tanx2=±1-cosx1+cosx;

here the sign (±) has to be chosen according to the quadrantMathworldPlanetmath where the angle x2 is.

Title derivation of half-angle formulae for tangent
Canonical name DerivationOfHalfangleFormulaeForTangent
Date of creation 2013-03-22 17:00:19
Last modified on 2013-03-22 17:00:19
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 9
Author rspuzio (6075)
Entry type Derivation
Classification msc 26A09
Related topic TangentOfHalvedAngle