# Trisecting an Angle Using a Lemniscate

Trisecting an Angle Using a Lemniscate

The angle to trisect, say , is between the longer leg of the carpenter's square (the brown "L" shape) and the polar axis . Translate the carpenter's square so that it touches the curve at the point . The angle between the radius vector and the polar axis is one-third of the given angle .

α

OA

P

AOP

OP

OA

α

For any curve in polar coordinates, the tangent of the angle between the tangent and radial line (the angle between vectors and ) at the point is . The lemniscate has the polar equation , and the derivative with respect to is , so . So . Since is obtuse, .

β

OP

PB

(r,θ)

tan(β)=r/r'

r=

cos(2θ)

θ

r'=

-sin(2θ)

cos(2θ)

r/r'=/=-cot(2θ)

cos(2θ

-sin(2θ)

cos(2θ)

tan(β)=-cot(2θ)=-tan(π/2-2θ)=tan(2θ-π/2)

β

β=2θ+π/2

So , and the angle between the tangent and the normal to the radius vector is . But this angle is equal to the angle between the larger leg of the carpenter's square and the radius vector , because these angles have orthogonal legs. So , .

∠OPB=π-β=π/2-2θ

BPC

PB

PC

OP

2θ

BPC

OP

α-θ=2θ

α=3θ