Question

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

Answer 1

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).