Find the exact value of all angles theta in the interval 0 2

Find the exact value of all angles theta in the interval [0, 2 pi) that satisfy the equations. Enter DNE in all capital letters into any unused answer boxes. (a) 2sin(2 theta) - 1 = 0 theta = (b) squareroot 3 csc(3 theta) - 2 = 0 theta = (c) csc(1/2 theta) - 1 = 0 theta = (d) squareroot 3 tan(2 theta) - 1 = 0 theta = (a) 2 cos^2 x + sin x = 2 x = (b) 2 cos^2 x - sin x = 1 x =

Solution

a)0<=<2

0<=2<4

2sin(2)-1=0

sin(2)=1_/2

2=_/6,5_/6,13_/6,17_/6

=_/12,5_/12,13_/12,17_/12

-------------------------------------

b)

0<=<2

0<=3<6

3 csc(3)-2=0

csc(3) =2_/3

3=_/3,2_/3,7_/3,8_/3,13_/3,14_/3,

=_/9,2_/9,7_/9,8_/9,13_/9,14_/9,

-------------------------------------

c)

0<=<2

0<=(1_/2)<

csc((1_/2)) -1=0

csc((1_/2)) =1

((1_/2)) =(_/2)

=

--------------------------

d)

0<=<2

0<=2<4

3 tan(2) -1=0

tan(2) =1_/3

2 =_/6,7_/6,13_/6,19_/6

=_/12,7_/12,13_/12,19_/12