Homework Statement

Suppose that an electric field is given by E(r,t)=E0cos(k·r−ωt+φ), where k⊥E0 and φ is a constant phase. Show that B(r,t)=((k×E)/ω)B(k⋅r-ωt+φ) is consistent with ∇×E=-∂B/∂t

∇×E=-∂B/∂t

The Attempt at a Solution

I know I have to take the curl of E, but I'm not sure how to go about doing it.
∂E/∂t would be ωE0sin(k⋅r-ωt+φ) and ∂E/∂r would be -kE0sin(k⋅r-ωt+φ), but I'm not sure if that helps. Also, by my calculation, -∂B/∂t= (k×E)cos(k⋅r-ωt+φ).
Any help would be greatly appreciated.

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