Area between two curves that don't intersect at a domain is
#int_a^bf(x)-g(x)#, where f(x) is the function with the larger y values. Either using a graphing calculator or some double angle identities, you find that the graphs intersect at #pi/6, pi/2, and (5pi)/6#. Because of the intersection at #pi/2# you'll have to split the area into two parts to use the formula above into: #[pi/6,pi/2]# and #[pi/2,(5pi)/6]#. But notice that in #[pi/6,pi/2]#, sin(2x)>cos(x), but its the opposite for the other domain, so make sure to put the larger function in the correct place into the formula:
I don't have the time right now to integrate this step by step (maybe someone else can?), although I'll tell you that #int_ ^ sin(2x)=-1/2cos(2x)#, and #int_ ^ cos(x)=sin(x)#. I hope you can carry on from there, it's just a lot of algebra (or plugging into calculator) You'll get -0.5, but area is positive, so it's just 0.5. Let me know if you have any questions.