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Find sin DAE in equilateral ABC, where D and E trisect BC

Source: AIME II 2013, Problem 5

April 4, 2013

trigonometrygeometrycircumcirclerationumber theoryrelatively primetrig identities

Problem Statement

In equilateral

\triangle ABC

let points

D

and

E

trisect

\overline{BC}

. Then

\sin \left( \angle DAE \right)

can be expressed in the form

\tfrac{a\sqrt{b}}{c}

, where

a

and

c

are relatively prime positive integers, and

b

is an integer that is not divisible by the square of any prime. Find

a+b+c