Sines And Cosines Reference Sheet (Page 2 of 5) in pdf

Sines And Cosines Reference Sheet Page 2

However, in practice, route 2) will only give a few values that cannot be found through route

1). There is however, another circumstance under which sin and cos of π/n can be found. If

n is a Fermat Prime i.e. a prime number of the form 2

+ 1, then sin π/n and cos π/n can be

found exactly. n = 3 and n = 5 are Fermat primes and the sine and cosine of π/3 and π/5 can

be found (see above).

The next Fermat prime is 17 and it turns out that sin π/17 and cos π/17 can be found. Simi-

larly, for the two remaining known Fermat primes n = 257 and n = 65537, the sine and cosine

of π/n can be found.

Using technique 1) above, on the basis of sinπ/17 and cos π/17, the sines and cosines of π/34,

π/68, π/136 etc. can be found.

Route 2) can be used for sin π/n and cos π/n where n is the product of two Fermat Primes

greater than 2.

Also, it is worth pointing out that if sin π/n and cos π/n can be found, then so can sin mπ/n

and cos mπ/n from the formulae for sin and cos of A + B etc.

So, sin π/n and cos π/n can be found for

n = 2

× 3

× 5

× 17

× 257

× 65537

where k is a positive or zero integer and the indices

p, q, r, s, t are all either zero or 1. Values of n under 1000 are as given in the following table.

102 120 128 136 160 170 192

204 240 255 256 257 272 320 340

384 408 480 510 512 514 544 640

680 768 771 816 960

The sines and cosines of values of mπ/n for n ≤ 32 are given in the following tables.

Sines And Cosines Reference Sheet Page 2