Question

evaluate 2 sin π/12​

Answer 1

To Find:

Evaluate 2 sin π/12.

Solution:

let, $P=\pi/6$

then,

$(sinP/2+cosP/2)^2=sin^2P/2+cos^2P/2+2sinP/2cosP/2\\\\(sinP/2+cosP/2)^2=1+sinP\\\\sinP/2+cosP/2=\sqrt{1+sinP}$ ...(1)

similarly,

$sinP/2-cosP/2=\pm\sqrt{1-sinP}$      

(as, for smaller value of P,  {P= π/12} , cos P > sin P ), therefore,

$sinP/2-cosP/2=-\sqrt{1-sinP}$        ....(2)  

on adding, equation (1) and (2), we get

$2 \ sinP/2=\sqrt{1+sinP}-\sqrt{1-sinP}$

on substituting, value of P , we get

$2sin\pi/12=\sqrt{1+sin\pi/6} -\sqrt{1-sin\pi/6}\\\\2sin\pi/12=\sqrt{3/2} -\sqrt{1/2}\\\\2sin\pi/12=\dfrac{\sqrt{3}-1}{\sqrt{2} }$

Required answer.