(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) Lets see it before doing slope any calculations a x of line is g(x) Lets 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! (cos (x+a), sin (x+a)) (cos x, sin x) Now lets 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.
Plane and Spherical Trigonometry With Applications (Boston: D. Heath and Co., c1943), by William Le Roy Hart (page images at HathiTrust); [Info]. Wentworth and David Eugene Smith, contrib. Wentworth (PDF at djm.cc); [Info]. The Elements Of Plane And Spherical Trigonometry.
(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 Thats what we wanted to show. X x+a-90 90-x-a/2 (cos (x+a), -sin (x+a)).