Math Editor

for the curve r = 1 + cosθ when is the tangent horizontal or c vertical

sol:

x=rcosθ→x=(1+cosθ)cosθy=rsinθ→y=(1+cosθ)sinθdθdy​=2cosθ2+cosθ−1dθdx​=−sinθ(2cosθ+1)dθdy​=0,dθdx​=0⟹2cosθ2+cosθ−1=0cosθ=u⟹2u2+u−1u=−1±√(1)2−4(2)(−1)​/2(2)u=−1±3/4⟹cosθ=21​,−1cosθ=−21​⟹32π​,34π​The tangent is horizontal at⟹3π​,π, 5π​The tangent is perpendicular to⟹32π​,π,34π​,2π​