angle multiplication and division formulae for tangent

From the angle addition formula for the tangentPlanetmathPlanetmathPlanetmath, we may derive formulae for tangents of multiples of angles:

tan(2x) =2tanx1-tan2x
tan(3x) =3tanx-tan3x1-3tan2x
tan(4x) =4tanx-3tan3x1-6tan2x+tan4x

These formulae may be derived from a recursion. Write tanx=w and write tan(nx)=un/vn where the u’s and the v’s are polynomials in w. Then we have the initial values u1=w and v1=1 and the recursions

un+1 =un+wvn
vn+1 =vn-wun,

which follow from the addition formulaPlanetmathPlanetmath. Moreover, if we know the tangent of an angle and are interested in finding the tangent of a multiple of that angle, we may use our recursions directly without first having to derive the multiple angle formulae. From these recursions, one may show that the u’s will only involve odd powers of w and the v’s will only involve even powers of w.

Proceeding in the opposite direction, one may consider bisecting an angle. Solving for tanx in the duplication formula above, one arrives at the following half-angle formula:


Expressing the tangent in terms of sines and cosines and simplifying, one finds the following equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath formulae:

Title angle multiplication and division formulae for tangent
Canonical name AngleMultiplicationAndDivisionFormulaeForTangent
Date of creation 2013-03-22 17:00:15
Last modified on 2013-03-22 17:00:15
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type Result
Classification msc 26A09