xa

(cos x, sin x)

(cos (x+a), sin (x+a))

xa

(cos x, sin x)

(cos (x+a), sin (x+a))

(cos (x+a), -sin (x+a))

slopeof line isg(x)

xa

(cos x, sin x)

(cos (x+a), sin (x+a))

(cos (x+a), -sin (x+a))

slopeof line isg(x)

Imagine rotatingangle x (with a staying constant)around the circle.

We need to showthat the anglebetween the dashed green lines is constant.

That shows the slope stays the same.

xa

(cos x, sin x)

(cos (x+a), sin (x+a))

(cos (x+a), -sin (x+a))

slopeof line isg(x)

Let’s see it before doingany calculations…

Let’s increase x just a little bit!

xa

(cos x, sin x)(cos (x+a), sin (x+a))

(cos (x+a), -sin (x+a))

old slope

new slope

Even with a larger x,we see the slope staysthe same!

xa

(cos x, sin x)

(cos (x+a), sin (x+a))

(cos (x+a), -sin (x+a))

slopeof line isg(x)

Now let’s prove that the angle betweenthe dashed green lines stays constant!

Otherwise known as “hello, isoceles triangles!”

xa

(cos x, sin x)

(cos (x+a), sin (x+a))

(cos (x+a), -sin (x+a))

180-x-a

180-x-a

90

90

x+a-90

x+a-90

xa

(cos x, sin x)

(cos (x+a), sin (x+a))

(cos (x+a), -sin (x+a))

180-x-a

180-x-a

x+a-90

x+a

90-x-a/2

90-x-a/2

xa

(cos x, sin x)

(cos (x+a), sin (x+a))

(cos (x+a), -sin (x+a))

x+a-9090-x-a/2

xa

(cos x, sin x)

(cos (x+a), sin (x+a))

(cos (x+a), -sin (x+a))

x+a-9090-x-a/2

The angle between the dashed green lines is:

(x+a-90)+(90-x-a/2)=a/2

which is not dependent on x at all.

That’s what we wanted to show.