Trigonometry Values Pi/3

From Trigonometry Values Pi/6

$\displaystyle \sin\left({\pi\over 6}\right)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}$	(1)
$\displaystyle \cos\left({\pi\over 6}\right)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}\sqrt{3}$	(2)

together with the trigonometric identity

$\begin{displaymath} \sin(2\alpha) = 2\sin\alpha\cos\alpha, \end{displaymath}$

(3)

the identity

$\begin{displaymath} \sin\left({\pi\over 3}\right)= 2\sin\left({\pi\over 6}\right... ...extstyle{1\over 2}}\sqrt{3}\,) ={\textstyle{1\over 2}}\sqrt{3} \end{displaymath}$

(4)

is obtained. Using the identity

$\begin{displaymath} \cos(2\alpha) = 1-2\sin^2\alpha, \end{displaymath}$

(5)

then gives

$\begin{displaymath} \cos\left({\pi\over 3}\right)= 1-2\sin^2\left({\pi\over 6}\right)= 1-2({\textstyle{1\over 2}})^2 = {\textstyle{1\over 2}}. \end{displaymath}$

(6)

Summarizing,

$\displaystyle \sin\left({\pi\over 3}\right)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}\sqrt{3}$	(7)
$\displaystyle \cos\left({\pi\over 3}\right)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}$	(8)
$\displaystyle \tan\left({\pi\over 3}\right)$	$\textstyle =$	$\displaystyle \sqrt{3}.$	(9)

See also Equilateral Triangle