Solve the following equation on the interval 0 2x cos x 2 s

Solve the following equation on the interval (0, 2x). cos x + 2 sin x cos x = 0 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. x = (Type an exact using x as needed Use a comma to separate answers as needed Type your answer in radians. Use integers or fractions for any numbers in the expression) There is no solution

Solution

cosx + 2sinx cosx = 0

==> cosx [1 + 2sinx] = 0

==> cosx = 0 ; 1 + 2sinx = 0

consider cosx = 0

==> x = /2 , 3/2 ; since the given interval is [0 , 2)

Consider 1 + 2sinx = 0

==> 2sinx = -1

==> sinx = -1/2

==> x = + (/6) ,2 - (/6) ; since sine is negative in third(Hence + /6) and fourth quadrants[Hence 2 - (/6)]

==> x = 7/6 , 11/6

There fore x = /2 , 7/6 ,3/2, 11/6