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