Question

24.
Find the general solution of the equation :
cos 3x + cos X - cos 2x = 0​

Answer 1

Answer:

cos 3x + cos x - cos 2x = 0

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

we knew that,

cos x + cos y = 2 cos (x + y /2) cos (x - y/2)

replacing x by 3x and y by x

2 cos ( 3x + x /2) . cos (3x - x/2) - cos2x = 0

2 cos(4x/2) . cos (2x/2) - cos 2x = 0

2 cos 2x . cos x - cos 2x = 0

cos 2x ( 2cos x -1 ) = 0

hence,

cos2x = 0

general solution for cos2x = 0

thus, general solution is

2x = (2n + 1)  π/2

x = (2n + 1) π/4

n ∈ z

(2cosx - 1) = 0

2 cos x = 1

cos x = 1/2

then, general solution for cos x = 1/2

we knew that cos 60 = 1/2

so,

60 = 60 x  π/180

    = π/3

let cos x = cos y

given cos x = 1/2

cos y = 1/2

cos y = cos(π/3)

y = π/3

general solution for cos x = cos y

x = 2nπ ± y , where n ∈ z

substituting y =π /3

x = 2nπ± π/3 , where n ∈ z

hence general solution for cos 2x = 0

x = (2n + 1 )π/4

        (or)

general solution for cos x = 1/2

x = 2nπ± π/3 , where n ∈ z