Tangent of 15 Degrees
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Theorem
- $\tan 15^\circ = \tan \dfrac {\pi} {12} = 2 - \sqrt 3$
where $\tan$ denotes tangent.
Proof 1
| \(\ds \tan 15^\circ\) | \(=\) | \(\ds \tan \frac {30^\circ} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sin 30^\circ} {1 + \cos 30^\circ}\) | Half Angle Formula for Tangent: Corollary 1 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\frac 1 2} {1 + \frac {\sqrt 3} 2}\) | Sine of $30^\circ$ and Cosine of $30^\circ$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\frac 1 2} {\frac {2 + \sqrt 3} 2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2 + \sqrt 3}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2 - \sqrt 3} {\left({2 + \sqrt 3}\right) \left({2 - \sqrt 3}\right)}\) | multiplying top and bottom by $2 - \sqrt 3$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2 - \sqrt 3} {4 - 3}\) | Difference of Two Squares | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 - \sqrt 3\) |
$\blacksquare$
Proof 2
| \(\ds \tan 15 \degrees\) | \(=\) | \(\ds \tan \frac {30 \degrees} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {1 - \cos 30 \degrees} {\sin 30 \degrees}\) | Half Angle Formula for Tangent: Corollary 2 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {1 - \frac {\sqrt 3} 2} {\frac 1 2}\) | Cosine of $30 \degrees$ and Sine of $30 \degrees$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 - \sqrt 3\) | multiplying top and bottom by $2$ |
$\blacksquare$
Proof 3
| \(\ds \tan 15^\circ\) | \(=\) | \(\ds \tan \frac {30^\circ} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sin 15^\circ} {\cos 15^\circ}\) | Tangent is Sine divided by Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\frac {\sqrt 6 - \sqrt 2} 4} {\frac {\sqrt 6 + \sqrt 2} 4}\) | Sine of $15^\circ$ and Cosine of $15^\circ$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sqrt 6 - \sqrt 2} {\sqrt 6 + \sqrt 2}\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\left({\sqrt 6 - \sqrt 2}\right)^2} {\left({\sqrt 6 + \sqrt 2}\right) \left({\sqrt 6 - \sqrt 2}\right)}\) | multiplying top and bottom by $\sqrt 6 - \sqrt 2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {6 - 2 \sqrt 6 \sqrt 2 + 2 } {6 - 2}\) | multiplying out, and Difference of Two Squares | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {8 - 4 \sqrt 3} 4\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 - \sqrt 3\) | dividing top and bottom by $4$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles