## Other angles related to

Look again at the sharp and flat triangles of the pentagon that we saw above. If we divide each in half, we have right angled triangles with sides 1 and Phi/2 around the 36° angle in the flat triangle and sides 1/2 and Phi around the 72° angle in the sharp triangle. So:

 cos (72°) = cos| '2 J = sin(18°) = sin ! 2 20 cos(36°) = cos (5) : I 2 2f

We have sin(18°) but what about cos(18°)? This has a somewhat more awkward expression as:

Now we know the sin and cos of both 30° and 18° we can find the sin and cos of their difference using:

and get:

AAAAgh! as Snoopy might have said.

Is there a neater (that is, a simpler) expression? Perhaps you can find one. Let me know if you do and it will be added here with your name!

This form of cos(12°) is derived from the expression on page 42 of

^^^ Roots of (H-L)/15 Recurrence Equations in Generalized Pascal Triangles by C Smith and V E Hoggatt Jr. in The Fibonacci Quarterly vol 18 (1980) pages 36-42.

What about other angles? From an equilateral triangle cut in half we can easily show that:

and from a 45-45-90 degree triangle we can derive:

1 V2

and not forgetting, of course:

Can you find any more angles that have an exact expression (not necessarily involving Phi or phi)? Let me know what you find and let's get a list of them here.