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Question
Find the general solution of the following equation:
sin θ = tan θ
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Solution
sin θ = tan θ
tan θ = `sin theta/cos theta`
sin θ = `sin theta/cos theta`
Multiply both sides by cos θ
sin θ ⋅ cos θ = sin θ
sin θ cos θ − sin θ = 0
⇒ sin θ(cos θ − 1) = 0
sin θ(cos θ − 1) = 0
Case 1: sin θ = 0
sinθ = 0 ⇒ θ = nπ, where n ∈ Z
Case 2: cosθ = 1
cos θ = 1
⇒ θ = 2nπ, where n ∈ Z
Consider excluded values (where cos θ = 0)
sin θ = `sintheta/costheta`
cos θ = 0
⇒ theta = `pi/2 + npi`, where n ∈ Z
θ = nπ, where n ∈ Z
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