angle multiplication and division formulae for tangent
AngleMultiplicationAndDivisionFormulaeForTangent
2007-04-27 23:33:54
2007-07-05 09:50:49
Result
addition and subtraction formulas for tangent
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From the angle addition formula for the tangent, we may derive formulae
for tangents of multiples of angles:
\begin{align*}
\tan (2x) &= {2 \tan x \over 1 - \tan^2 x} \\
\tan (3x) &= {3 \tan x - \tan^3 x \over 1 - 3 \tan^2 x} \\
\tan (4x) &= {4 \tan x - 3 \tan^3 x \over 1 - 6 \tan^2 x + \tan^4 x}
\end{align*}
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
\begin{align*}
u_{n+1} &= u_n + w v_n \\
v_{n+1} &= v_n - w u_n ,
\end{align*}
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:
\[
\tan \left( {x \over 2} \right) =
{1 - \cos x \over \sin x} =
{\sin x \over 1 + \cos x} =
\pm\sqrt{ 1 - \cos x \over 1 + \cos x }
\]