Find the number of solutions of the equation sin2θ+cos2θ+4sinθ=1+4cosθ lying in the interval [−2π,2π].
2sinθcosθ+1−2sin2θ+4sinθ=1+4cosθ
⇒2sinθ(cosθ−sinθ)−4(cosθ−sinθ)=0
⇒(2sinθ−4)(cosθ−sinθ)=0
But sinθ≠2
∴tanθ=1⇒θ=nπ+π4,nϵI
∴ Number of solutions in [−2π,2π] are 4,i.e.,θ=−7π4−3π4,π4,5π4