What is the equation of the tangent line of r=cos(θ+π2)⋅cos(θ−π) at θ=−π8?

1 Answer
namralos
Mar 25, 2017

Simplify before you differentiate.

Explanation:

By the Cosine Sum Formula
cos(θ+π2)=cos(θ)cos(π2)−sin(θ)sin(π2)=0−sin(θ)=−sin(θ)
By the Cosine Difference Formula
cos(θ−π)=cos(θ)cos(π)+sin(θ)sin(π)=−cos(θ)+0=−cos(θ)
Their product is
r=(−sin(θ))(−cos(θ))=sin(θ)cos(θ)
By the Sine Double angle formula, we have
r=sin(θ)cos(θ)=(12)sin(2θ)

To find the slope of the tangent line, use the fact that

dydx=drd(θ)sin(θ)+rcos(θ)drd(θ)cos(θ)−rsin(θ)

Finally, determine the point (x, y) corresponding to the point that you are interested in, where θ=−π8. Hint: x=rcos(θ)andy=rsin(θ). Once you have a point and the slope, use the Point-Slope Form of a line to obtain its equation.

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