Find the slope of the tangent line to the polar curve r = \sin (6 \theta) at \theta = \frac{\pi}{12}.

By converting into parametric equations,

{x(θ)=r(θ)cosθ=cos2θcosθy(θ)=r(θ)sinθ=cos2θsinθ

By Product Rule,

x'(θ)=−sin2θcosθ−cos2θsinθ

x'(π2)=−sin(π)cos(π2)−cos(π)sin(π2)=1

y'(θ)=−sin2θsinθ+cos2θcosθ

y'(π2)=−sin(π)sin(π2)+cos(π)cos(π2)=0

So, the slope m of the curve can be found by

m=dydx∣θ=π2=y'(π2)x'(π2)=01=0

I hope that this was helpful.