Question

An equation is given. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.) 2 cos(2θ) − 1 = 0 (a) Find all solutions of the equation. θ = (b) Find the solutions in the interval [0, 2π).

Answer 1

Answer:

a) The general solution   θ = nπ±π/6

The all solutions are  - π/6 ,π/6 ,5π/6 ,7π/6 ,11π/6 ,13π/6 ...…

b)   The solutions in the interval [ 0,2π)

                            θ =  - π/6 ,π/6 , 5π/6 , 7π/6 , 11π/6

Step-by-step explanation:

Step( i):-

Given an equation 2 cos (2θ)-1 =0

                               2 cos (2θ) = 1

                                cos(2θ) = 1/2

                                cos(2θ) = cos (π/3)

Step(ii):-

a)

The general solution of  cos θ = cos ∝ is given by

                                              θ = 2nπ±∝

The general solution of  cos(2θ) = cos (π/3) is

                                               2θ = 2nπ±π/3

                                                θ = nπ±π/6

Put n=0          θ =  ±π/6

Put n =1          θ = π±π/6

                       θ = π-π/6 = 5π/6

                        θ = π+π/6 = 7π/6

put n =2

                        θ = 2π±π/6  

                         θ = 2π-π/6 = 11π/6

                        θ = 2π+π/6 = 13π/6

   

The all solutions are  - π/6 ,π/6 ,5π/6 ,7π/6 ,11π/6 ,13π/6 ...…

b)

        The solutions in the interval [ 0,2π)

                            θ =  - π/6 ,π/6 , 5π/6 , 7π/6 , 11π/6

Final answer:-

a)

The all solutions are  - π/6 ,π/6 ,5π/6 ,7π/6 ,11π/6 ,13π/6 ...…

b)

The solutions in the interval [ 0,2π)

                            θ =  - π/6 ,π/6 , 5π/6 , 7π/6 , 11π/6