Find all exact values at x in the interval 0 2pi that satisf

Find all exact values at x in the interval [0, 2pi] that satisfy each equation. 2 sin (x) -1 = 0 2 cos (3pi) + Squareroot2 = 0.

Solution

a) 2sin(x) - 1 = 0

==> 2sinx = 1

==> sinx = 1/2

==> x = /6 , 5/6 (since sine function is positive in both first and second quadrants)

b) 2cos(3x) + 2 = 0

==> 2cos(3x) = -2

==> cos(3x) = - (2)/2

==> cos(3x) = -1/2

==> 3x = 3/4 , 5/4 , 11/4 , 13/4 , 19/4 , 21/4

==> x = /4 , 5/12 , 11/12 , 13/12 , 19/12 , 21/12

Hence x = /4 , 5/12 , 11/12 , 13/12 , 19/12 , 7/4