Question

Solve the following equations:
sin 5x − sin x = cos 3

Answer 1

`2cos ((5x + x)/2) * sin((5x - x)/2)` = cos 3x

`2 cos (6x/2) * sin (4x/2)` = cos 3x

2 cos 3 x . sin 2x = cos 3x

2 cos 3x . sin 2x – cos 3x = 0

cos 3x (2 sin 2x – 1) = 0

cos 3x = 0 or 2 sin 2x – 1 = 0

cos 3x = 0 or sin 2x = `1/2`

To find the general solution of cos 3x = 0

The general solution of cos 3x = 0 is

3x = `(2"n" + 1)^(pi/2)`, n ∈ Z

x = `(2"n" + 1)^(pi/6)`, n ∈ Z

To find the general solution of sin 2x = `1/2`

sin 2x = `1/2`

sin 2x = `sin (pi/6)`

The general solution is

2x = `"n"pi + (- 1)^"n"  pi/6`, n ∈ Z

x = `("n"pi)/2 + (- 1)^"n"  pi/12`, n ∈ Z

∴ The required solutions are

x = `(2"n" + 1)  pi/6`, n ∈ Z

x = `("n"pi)/2 + (- 1)^"n"  pi/12`, n ∈ Z