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- 1. (cos (x+a), sin (x+a)) (cos x, sin x) a x
- 2. (cos (x+a), sin (x+a)) (cos x, sin x) slope a x of line is g(x) (cos (x+a), -sin (x+a))
- 3. (cos (x+a), sin (x+a)) Imagine rotating angle x (with a staying constant) (cos x, sin x) around the circle. We need to show slope that the angle a x of line is between the g(x) dashed green lines is constant. That shows the slope stays the same. (cos (x+a), -sin (x+a))
- 4. (cos (x+a), sin (x+a)) (cos x, sin x) Let’s see it before doing slope any calculations… a x of line is g(x) Let’s increase x just a little bit! (cos (x+a), -sin (x+a))
- 5. (cos (x+a), sin (x+a)) (cos x, sin x) new slope a old slope x Even with a larger x, (cos (x+a), -sin (x+a)) we see the slope stays the same!
- 6. (cos (x+a), sin (x+a)) (cos x, sin x) Now let’s prove that the angle between slope the dashed green lines stays constant!line is a x of g(x) Otherwise known as “hello, isoceles triangles!” (cos (x+a), -sin (x+a))
- 7. (cos (x+a), sin (x+a)) 180-x-a x+a-90 (cos x, sin x) 90 a x 90 180-x-a x+a-90 (cos (x+a), -sin (x+a))
- 8. (cos (x+a), sin (x+a)) 180-x-a (cos x, sin x) a x 90-x-a/2 180-x-a x+a x+a-90 90-x-a/2 (cos (x+a), -sin (x+a))
- 9. (cos (x+a), sin (x+a)) (cos x, sin x) a x x+a-90 90-x-a/2 (cos (x+a), -sin (x+a))
- 10. (cos (x+a), sin (x+a)) The angle between the dashed green lines is: (x+a-90)+(90-x-a/2)=a/2 (cos x, sin x) which is not dependent on x at all. a That’s what we wanted to show. x x+a-90 90-x-a/2 (cos (x+a), -sin (x+a))

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