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