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
\(\ds \tan 15 \degrees\) | \(=\) | \(\ds \tan \frac {30 \degrees} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin 30 \degrees} {1 + \cos 30 \degrees}\) | Half Angle Formula for Tangent: Corollary $1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\frac 1 2} {1 + \frac {\sqrt 3} 2}\) | Sine of $30 \degrees$ and Cosine of $30 \degrees$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\frac 1 2} {\frac {2 + \sqrt 3} 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 + \sqrt 3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 - \sqrt 3} {\paren {2 + \sqrt 3} \paren {2 - \sqrt 3} }\) | 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
\(\ds \tan 15 \degrees\) | \(=\) | \(\ds \frac {\sin 15 \degrees} {\cos 15 \degrees}\) | Tangent is Sine divided by Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\frac {\sqrt 6 - \sqrt 2} 4} {\frac {\sqrt 6 + \sqrt 2} 4}\) | Sine of $15 \degrees$ and Cosine of $15 \degrees$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sqrt 6 - \sqrt 2} {\sqrt 6 + \sqrt 2}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {\sqrt 6 - \sqrt 2}^2} {\paren {\sqrt 6 + \sqrt 2} \paren {\sqrt 6 - \sqrt 2} }\) | 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