Best way to calculate coordinates?

[MathFan]

Hi,

If a regular pentagon is inscribed in a circle, what is the quickest (most efficient) way to calculate the coordinates of the vertices given that the centre of the circle is the origin and one of the vertices is on the +ve x-axis?

For example...


Many thanks for any input.

 

[soroban]

Hello, MathFan!

If a regular pentagon is inscribed in a circle,
what is the quickest (most efficient) way to calculate the coordinates
of the vertices given that the centre of the circle is the origin
and one of the vertices is on the positive x-axis?

For example:

View attachment 1615

Call that first vertex \(\displaystyle A\).
Label the vertices counterclockwise: \(\displaystyle B,C,D,E.\)

Then we have: .\(\displaystyle \begin{Bmatrix}
A & (5\cos0^o,5\sin0^o) &\approx& (5, 0) \\
B & (5\cos72^o,5\sin72^o) & \approx & (1.545, 4.755) \\
C & (5\cos144^o,5\sin144^o) & \approx & (\text{-}4.045, 2.939)\\
D & (5\cos216^o,5\sin216^o) & \approx & (\text{-}4,045, \text{-}2.939) \\
E & (5\cos288^o,5\sin288^o) & \approx & (1.545, \text{-}4.755)
\end{Bmatrix}\)

 

[pka]

If a regular pentagon is inscribed in a circle, what is the quickest (most efficient) way to calculate the coordinates of the vertices given that the centre of the circle is the origin and one of the vertices is on the +ve x-axis?

This is the same answer given in reply #2. However, if working with a programmable calculator of computer algebra it can be generalized.
\(\displaystyle \left( {5\cos \left( {\frac{{2k\pi }}{5}} \right),5\sin \left( {\frac{{2k\pi }}{5}} \right)} \right),\;k = 0, \cdots, 4\)

Generalized: if you have a regular N-sided polygon the change 5 to N and 4 to N-1.

 

[MathFan]

Many thanks to both of you for your help ;-)