|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
angle multiplication and division formulae for tangent
|
(Result)
|
|
|
From the angle addition formula for the tangent, we may derive formulae for tangents of multiples of angles:
These formulae may be derived from a recursion. Write $\tan x = w$ and write $\tan (nx) = u_n / v_n$ where the $u$ 's and the $v$ 's are polynomials in $w$ . Then we have the initial values $u_1 = w$ and $v_1 = 1$ and the recursions
which follow from the addition formula. 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 $\tan x$ in the duplication formula above, one arrives at the following half-angle formula: $$ \tan \left( {x \over 2} \right) = \sqrt{ 1 + {1 \over \tan^2 x}} - {1 \over \tan x} $$ Expressing the tangent in terms of sines and cosines and simplifying, one finds the following equivalent formulae: $$ \tan \left( {x \over 2} \right) = {1 - \cos x \over \sin x} = {\sin x \over 1 + \cos x} = \pm\sqrt{ 1 - \cos x \over 1 + \cos x } $$
|
"angle multiplication and division formulae for tangent" is owned by rspuzio. [ full author list (3) ]
|
|
(view preamble | get metadata)
Cross-references: equivalent, cosines, sines, terms, formula, opposite, even powers, odd powers, polynomials, multiples, tangent, addition formula, angle
This is version 5 of angle multiplication and division formulae for tangent, born on 2007-04-27, modified 2007-07-05.
Object id is 9286, canonical name is AngleMultiplicationAndDivisionFormulaeForTangent.
Accessed 1628 times total.
Classification:
| AMS MSC: | 26A09 (Real functions :: Functions of one variable :: Elementary functions) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
