The expression √3·sinθ + cosθ is rewritten in the form R·sin(θ + α) with R > 0 and α acute. Then:

Expanding, $R$·sin(θ + α) = $(R$·cosα)sinθ + $(R$·sinα)cosθ, so $R$·cosα = √3 and $R$·sinα = 1. Squaring and adding gives $R^2=4$, i.e. $R=2$; dividing gives tanα = 1/√3, i.e. α = 30°. Check at θ = 60°: √3(√3/2) + 1/2 = 2 = 2·sin90°. Swapping which equation supplies the tangent gives α = 60°, forgetting the square root gives $R=4$, and simply adding the coefficients gives $R$ = √3 + 1.