Triple Angle Formulas/Tangent
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Theorem
- $\tan 3 \theta = \dfrac {3 \tan \theta - \tan^3 \theta} {1 - 3 \tan^2 \theta}$
where $\tan$ denotes tangent.
Corollary 1
- $\tan 3 \theta = \tan \theta \dfrac {4 \cos^2 \theta - 1} {4 \cos^2 \theta - 3}$
Proof 1
\(\ds \tan 3 \theta\) | \(=\) | \(\ds \frac {\sin 3 \theta} {\cos 3 \theta}\) | Tangent is Sine divided by Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 \sin \theta - 4 \sin^3 \theta} {4 \cos^3 \theta - 3 \cos \theta}\) | Triple Angle Formula for Sine and Triple Angle Formula for Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 \sin \theta - 4 \sin^3 \theta} {4 \cos^3 \theta - 3 \cos \theta}\frac {\cos^3 \theta} {\cos^3 \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\frac {3 \tan \theta} {\cos^2 \theta} - 4 \tan^3 \theta} {4 - \frac {3 \cos \theta} {\cos^3 \theta} }\) | Tangent is Sine divided by Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 \tan \theta \sec^2 \theta - 4 \tan^3 \theta} {4 - 3 \sec^2 \theta}\) | Secant is Reciprocal of Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 \tan \theta \paren {1 + \tan^2 \theta} - 4 \tan^3 \theta} {4 - 3 \paren {1 + \tan^2 \theta} }\) | Difference of Squares of Secant and Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 \tan \theta + 3 \tan^3 \theta - 4 \tan^3 \theta} {4 - 3 - 3 \tan^2 \theta}\) | multiplying out | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 \tan \theta - \tan^3 \theta} {1 - 3 \tan^2 \theta}\) | gathering terms |
$\blacksquare$
Proof 2
Let $\theta$ be such that $\tan 2 \theta$ is defined.
Then:
\(\ds \tan 3 \theta\) | \(=\) | \(\ds \dfrac {\tan \theta + \tan 2 \theta} {1 - \tan \theta \tan 2 \theta}\) | Tangent of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\tan \theta + \dfrac {2 \tan \theta} {1 - \tan^2 \theta} } {1 - \tan \theta \dfrac {2 \tan \theta} {1 - \tan^2 \theta} }\) | Double Angle Formula for Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\tan \theta \paren {1 - \tan^2 \theta} + 2 \tan \theta} {\paren {1 - \tan^2 \theta} - 2 \tan^2 \theta}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\tan \theta - \tan^3 \theta + 2 \tan \theta} {1 - \tan^2 \theta - 2 \tan^2 \theta}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {3 \tan \theta - \tan^3 \theta} {1 - 3 \tan^2 \theta}\) | simplifying |
$\Box$
Now suppose $\theta$ is such that $\tan 2 \theta$ is not defined.
Then:
- $2 \theta = \dfrac \pi 2 + n \pi$
for some integer $n$.
Hence:
- $\theta = \dfrac \pi 4 + \dfrac {n \pi} 2$
For $n$ even we will then have:
- $\tan \theta = 1$
and:
- $\tan 3 \theta = -1$
For $n$ odd we will then have:
- $\tan \theta = -1$
and:
- $\tan 3 \theta = 1$
It is then directly verified that the Triple Angle Formula for Tangent holds for these special cases where $\tan 2 \theta$ is undefined.
$\blacksquare$
Proof 3
From Tangent of Sum of Three Angles:
- $\map \tan {A + B + C} = \dfrac {\tan A + \tan B + \tan C - \tan A \tan B \tan C} {1 - \tan B \tan C - \tan C \tan A - \tan A \tan B}$
The result follows by setting $\theta = A = B = C$.
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.46$