Question

solve the equation sin(2 theata +pi/3)=1/2 for 0<=theata<=pi,giving theta in terms of pi in your answers.

Answer 1

sin(2Ø + π/3) = 1/2
we know sin (π/6) = 1/2 , (sin 30° = 1/2)

so 2Ø + π/3 = π/6 OR 2Ø + π/3 = 5π/6
(sin30° = 1/2 and sin150° = 1/2)

case1:
2Ø + π/3 = π/6
2Ø = π/6 - π/3 = -π/6
Ø = -π/12 ---> -15°
but the period of sin(2Ø + π/3) is π
so adding multiples of π to our first answer will yield another answer
Ø = -π/12+π = 11π/12 ---> 165°
Ø = 11π/12 + π = 23π/12 ---> 345°

case2:
2Ø + π/3 = 5π/6
2Ø = 5π/6 - π/3 = π/2
Ø = π/4 ---> 45°
adding multiples of π
Ø = π/4+π = 5π/4 ---> 225°
Ø = 5π/4+π = 9π/4 , which is too large

Ø = π/4, 11π/12, 5π/4 , 23π/12

see:
http://www.wolframalpha.com/input/?i=plot+sin(2x+%2Bpi%2F3)%3D1%2F2+from+0+to+2%CF%80