Tangent of 75 Degrees
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Theorem
- $\tan 75 \degrees = \tan \dfrac {5 \pi} {12} = 2 + \sqrt 3$
where $\tan$ denotes tangent.
Proof
| \(\ds \tan 75 \degrees\) | \(=\) | \(\ds \frac {\sin 75 \degrees} {\cos 75 \degrees}\) | Tangent is Sine divided by Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\frac {\sqrt 6 + \sqrt 2} 4} {\frac {\sqrt 6 - \sqrt 2} 4}\) | Sine of $75 \degrees$ and Cosine of $75 \degrees$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sqrt 6 + \sqrt 2} {\sqrt 6 - \sqrt 2}\) | 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