How do you find the value of #sin((5pi)/12)#?

1 Answer
Jul 23, 2016

Actually it is very easy to find that #color(blue){sin({5\pi}/12)={sqrt(6)+sqrt(2))/4#.

Explanation:

You use the formula for the sine of a sum:

#sin(a+b)=sin a cos b+cos a sin b#

Think of a pie divided into twelve slices. Three of them form one quarter, two more form one sixth of the pie. So, thinking of the "pie" as #\pi#:

#{5 pi}/{12}={pi}/4+{pi}/6#.

With that in mind, put #a=\pi/4, b=\pi/6# into the sine formula given above:

#sin({5 pi}/12)=sin (pi/4) cos (pi/6)+cos (pi/4) sin (pi/6)#

Plug in the familiar values #sin(pi/4)=cos(pi/4)={sqrt(2)}/2#,#cos(pi/6)={sqrt(3)}/2#,#sin(pi/6)=1/2#, and you get the answer given above. And, yes, putting that expression into a calculator gives #0.9659# (four significant digits).