Trigonometry Pdf
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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.