Tangent of 30 Degrees

Theorem

$\tan 30 \degrees = \tan \dfrac \pi 6 = \dfrac {\sqrt 3} 3$

where $\tan$ denotes tangent.

$\ds \tan 30 \degrees$

$=$

$\ds \frac {\sin 30 \degrees} {\cos 30 \degrees}$

$\ds $

$=$

$\ds \frac {\frac 1 2} {\frac {\sqrt 3} 2}$

$\ds $

$=$

$\ds \frac 1 {\sqrt 3}$

$\ds $

$=$

$\ds \frac {\sqrt 3} 3$

multiplying top and bottom by $\sqrt 3$

$\blacksquare$

Some sources present the as:

$\tan 30 \degrees = \tan \dfrac \pi 6 = \dfrac 1 {\sqrt 3}$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): trigonometric function
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): trigonometric function
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Trigonometric values for some special angles
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Trigonometric values for some special angles