Pentagon by trigonometry

Question

Incredibly simple thing that I am having trouble with for the past hour or so.

With a pentagon like so http://prntscr.com/hx93vt

how do you work out the coordinates of the point c? I just don't understand how you are meant to figure it out not matter how many triangles I draw I don't seem to reach the right answer (Cos(4/5 pie),Sin(4/5pie))

(theta is 2/5 pie)

P.s sorry about the image I ran out of paper had to use paper plates lol

Answer 1

Whole angle is $2\pi$. it should be divided for 5 equal parts for $\frac{2\pi}{5}$ each. So point

$A(\cos \frac{2\pi\cdot 0}{5}; \sin \frac{2\pi\cdot 0}{5})$,

$B(\cos \frac{2\pi\cdot 1}{5}; \sin \frac{2\pi\cdot 1}{5})$,

$C(\cos \frac{2\pi\cdot 2}{5}; \sin \frac{2\pi\cdot 2}{5})$,

$D(\cos \frac{2\pi\cdot 3}{5}; \sin \frac{2\pi\cdot 3}{5})$,

$E(\cos \frac{2\pi\cdot 4}{5}; \sin \frac{2\pi\cdot 4}{5})$.

Answer 2

The center of the pentagon is $O = (0,0)$ and the vertices are $A_0,A_1,A_2,A_3,A_4$ and the radius is $1$

So the angles $\angle A_iOA_{i-1}= \frac {2\pi }5$.

And the angle for $\angle A_iOA_) = i*\frac {2\pi} 5$

So the cordinates of $A_i = (\cos (i*\frac {2\pi}5), \sin (i*\frac {2\pi}5))$.