How do you prove $sin 5pi/12 = sin pi/6 cos pi/4 + cos pi/6 sin pi/4$ ?

Question

4337 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-05-25T21:55:12

Note that $pi/6+pi/4=(2pi)/12+(3pi)/12=(5pi)/12$ .

Also, recognize the sine addition formula:

$sin(A+B)=sin(A)cos(B)+cos(A)sin(B)$

So, we can write $(5pi)/12$ as $pi/6+pi/4$ and then apply the sine addition formula.

$sin((5pi)/12)=sin(pi/6+pi/4)=sin(pi/6)cos(pi/4)+cos(pi/6)sin(pi/4)$

How do you prove sin 5pi/12 = sin pi/6 cos pi/4 + cos pi/6 sin pi/4sin 5pi/12 = sin pi/6 cos pi/4 + cos pi/6 sin pi/4?