Descartes's Angle Trisection

a = sin(3t)

0.75

zoom

1

2

Descartes used the intersection of a circle and a parabola to trisect an angle. The equations of the circle and parabola are

2

x

+

2

y

-

13

4

x+4ay=0

and

2

y

=

1

4

x

, respectively. The coordinates of the intersection satisfy

4

3

y

=3y-a

. Since

4

3

sin

(t)=3sin(t)-sin(3t)

, by taking

a=sin(3t)

, the smaller positive root of the last equation is

sin(t)

.