What is the equation of the line that is normal to the polar curve f(θ)=sin(2θ+π)θ at θ=π4?

1 Answer
May 19, 2017

(y42π28)=(π8+π)(x42π28)

Explanation:

Find r when θ=π4
r=sin(2(π4)+π)θ=4π4

Find drdθ:
r=sin(2θ+π)θ
drdθ=cos(2θ+π)(2)1=2cos(2θ+π)1

Substitute θ=π4:
=2cos(2(π4)+π)1=1

Find the derivative:
dydx=(1)sin(π4)+(4π4)cos(π4)(1)cos(π4)(4π4)sin(π4)=8π1

Find the respective x and y.
x=rcos(θ)=(4π4)(22)=42π28

y=rsin(θ)=(4π4)(22)=42π28

The slope of a normal line is the negative reciprocal of the derivative.
18π1=π8+π

Putting it all together:
(yy1)=m(xx1)
(y42π28)=(π8+π)(x42π28)