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Question

# The general value of x satisfying the equation √3 sin x+cos x=√3 is given by

A

x=nπ+(1)n π4+π3,nZ

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B

x=nπ+(1)n π3+π6,nZ

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C

x=nπ± π6,nZ

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D

x=nπ± π3,nZ

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Solution

## The correct option is B x=nπ+(−1)n π3+π6,n∈Z Given:√3 sin x+cos x=√3 ...(i)This equation is of the form a sinθ+b cosθ=c ,where a=√3b=1 and c=√3Let:a=r cos α and b=r sin αNow,r=√a2+b2=√(√3)2+12=2 and tan α=ba⇒tan α=1√3⇒ α =π6On puting α =√3=r cos α andb=1=r sin α in equation (i) ,we get:r cos α sin x+r sin α cos x=√3⇒r sin (x+α)=√3⇒ 2 sin (x+α)=√3⇒ sin (x+α)=√32⇒ sin (x+α)=sinπ3⇒ sin (x+π6)=sin π3⇒ x=nπ+(−1)n π3−π6,n∈Z

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