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Question
Solve the following equations:
sin θ + sin 3θ + sin 5θ = 0
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Solution
sin θ + sin 3θ + sin 5θ = 0
`2 sin ((5theta + theta)/2) * cos ((5theta - theta)/2) + sin 3theta` = 0
`2sin ((6theta)/2) * cos ((4theta)/2) + sin 3theta` = 0
2 sin 3θ . cos 2θ + sin 3θ = 0
sin 3θ (2 cos 2θ + 1) = θ
sin 3θ = 0 or 2 cos 2θ + 1 = θ
sin 3θ = 0 or cos 2θ = `- 1/2`
To find the general solution of sin 3θ = 0
The general solution is
3θ = nπ, n ∈ Z
θ = `("n"pi)/3`, n ∈ Z
To find the general solution of cos 2θ = ` - 1/2`
cos 2θ = ` - 1/2`
cos 2θ = `cos (pi - pi/3)`
cos 2θ = `cos ((3pi - pi)/3)`
cos 2θ = `cos ((2pi)/3)`
The general solution is
2θ = `2"n"pi +- (2pi)/3`, n ∈ Z
θ = `"n"pi +- pi/3`, n ∈ Z
∴ The required solutions are
θ = `(:"n"pi)/3`, n ∈ Z
or
θ = `"n"pi +- pi/3`, n ∈ Z
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