Cosines of multiples of π/34 in radical form.
Version of Thursday 25 February 2016.
Dave Barber's other pages.

A famous mathematical result attributed to C.F. Gauss is that the regular polygon of 17 sides is constructible using compass and straightedge.

Because these two tools are also enough to bisect any angle, the regular polygon of 34 sides is similarly constructible. It follows that cosines and sines of multiples of π/34 of can be written using nested square roots. Although the formulas are unwieldy if written entirely in primitives, the matter is streamlined if the constants of table one are defined:

table one
a= √17≈ 4.123106
b= √(34 + 2a)≈ 6.499709
c= √(34 − 2a)≈ 5.074819
d= √(17 + 3a − 2bc)≈ 3.360815
e= a/8 + c/8 + d/4 − 1/8 ≈ 1.864944
f= √(2 − e)≈ 0.367499
g= √(2 + e)≈ 1.965946

whence:

table two
cos ( 1 π/34)= 12 fg (− 2e + e3)≈ 0.995734
cos ( 2 π/34)= 12 g≈ 0.982973
cos ( 3 π/34)= 12 f (−1 − 2e + e2 + e3)≈ 0.961826
cos ( 4 π/34)= 12 e≈ 0.932472

cos ( 5 π/34)= 12 fg (−1 + e2)≈ 0.895163
cos ( 6 π/34)= 12 g (−1 + e)≈ 0.850217
cos ( 7 π/34)= 12 f (−1 + e + e2)≈ 0.798017
cos ( 8 π/34)= 12 (−2 + e2)≈ 0.739009

cos ( 9 π/34)= 12 fge≈ 0.673696
cos (10 π/34)= 12 g (−1 − e + e2)≈ 0.602635
cos (11 π/34)= 12 f (1 + e)≈ 0.526432
cos (12 π/34)= 12 (− 3e + e3)≈ 0.445738

cos (13 π/34)= 12 fg≈ 0.361242
cos (14 π/34)= 12 g (1 − 2 ee2 + e3)≈ 0.273663
cos (15 π/34)= 12 f≈ 0.183750
cos (16 π/34)= 12 (2 − 4e2 + e4)≈ 0.092268

These formulas are based on those at MathWorld, but have been extensively rearranged by the present author.

Each cosine is here written as a polynomial in e, f, and g; thus explicit radicals are gone and further calculations might be eased. In particular, the tangent function, being the ratio of a sine and a cosine, gives the appearance of being a rational function.

As always, sin (x) = cos (π/2 − x).

Squares of cosines are sometimes useful. In this representation, f and g drop out:

table three
cos2 ( 1 π/34)= 14 (16e2 − 20e4 + 8e6e8)≈ 0.991487
cos2 ( 2 π/34)= 14 (2 + e)≈ 0.966236
cos2 ( 3 π/34)= 14 (2 + 7e − 14e3 + 7e5e7)≈ 0.925109
cos2 ( 4 π/34)= 14 (e2)≈ 0.869504

cos2 ( 5 π/34)= 14 (4 − 9e2 + 6e4e6)≈ 0.801317
cos2 ( 6 π/34)= 14 (2 − 3e + e3)≈ 0.722869
cos2 ( 7 π/34)= 14 (2 − 5e + 5e3e5)≈ 0.636831
cos2 ( 8 π/34)= 14 (4 − 4e2 + e4)≈ 0.546134

cos2 ( 9 π/34)= 14 (4e2e4)≈ 0.453866
cos2 (10 π/34)= 14 (2 + 5e − 5e3 + e5)≈ 0.363169
cos2 (11 π/34)= 14 (2 + 3ee3)≈ 0.277131
cos2 (12 π/34)= 14 (9e2 − 6e4 + e6)≈ 0.198683

cos2 (13 π/34)= 14 (4 − e2)≈ 0.130496
cos2 (14 π/34)= 14 (2 − 7e + 14e3 − 7e5 + e7)≈ 0.074891
cos2 (15 π/34)= 14 (2 − e)≈ 0.033764
cos2 (16 π/34)= 14 (4 − 16e2 + 20e4 − 8e6 + e8)≈ 0.008513

Each of the above polynomials in e has a tractable factorization. Here are some breakdowns:

table four
3 π/34:  2 + 7e − 14e3 + 7e5e7 = (2 sin (1 π/14) + e)2 (2 sin (3 π/14) − e)2 (2 sin (5 π/14) + e)2 (2 sin (7 π/14) − e)
14 π/34: 2 − 7e + 14e3 − 7e5 + e7 = (2 sin (1 π/14) − e)2 (2 sin (3 π/14) + e)2 (2 sin (5 π/14) − e)2 (2 sin (7 π/14) + e)

7 π/34: 2 − 5e + 5e3e5 = (2 sin (1 π/10) − e)2 (2 sin (3 π/10) + e)2 (2 sin (5 π/10) − e)
10 π/34: 2 + 5e − 5e3 + e5 = (2 sin (1 π/10) + e)2 (2 sin (3 π/10) − e)2 (2 sin (5 π/10) + e)

11 π/34: 2 + 3ee3 = (2 sin (1 π/6) + e)2 (2 sin (3 π/6) − e)
6 π/34: 2 − 3e + e3 = (2 sin (1 π/6) − e)2 (2 sin (3 π/6) + e)

16 π/34: 4 − 16e2 + 20e4 − 8e6 + e8 = (2 + √2 − e2)2 (2 − √2 − e2)2
1 π/34: 16e2 − 20e4 + 8e6e8 = e2 (2 − e2)2 (4 − e2)

5 π/34: 4 − 9e2 + 6e4e6 = (1 − e2)2 (4 − e2)
12 π/34: 9e2 − 6e4 + e6 = e2 (3 − e2)2

8 π/34: 4 − 4e2 + e4 = (2 − e2)2
9 π/34: 4e2e4 = e2 (4 − e2)

The Chebyshev polynomial of the first kind, of degree 17, is:

T17 (x) = + 17x1 − 816x3 + 11424x5 − 71808x7 + 239360x9 − 452608x11 + 487424x13 − 278528x15 + 65536x17

which evaluates as follows:

• T17 (cos (n π/34)) = 0 if n is an odd integer
• T17 (cos (m π/17)) = −1 if m is an odd integer
• T17 (cos (m π/17)) = +1 if m is an even integer

These cosines can be used to found a variation on the quintright integral domain. An ordinary quintright, which has four integer components, is based on dividing a right angle into five equal parts (18 degrees). The version developed here, named QR17, divides the right angle into seventeen equal parts (5517 degrees).

Each QR17 is an ordered 16-tuple of integers whose components are subscripted 1 through 16. A QR17 is written as a comma-separated list between shallow angle brackets, except every fourth comma is replaced by a semicolon for ease of reading. If Q is a QR17, then its notation is:

Q ≡ ⟨ Q1, Q2, Q3, Q4; Q5, Q6, Q7, Q8; Q9, Q10, Q11, Q12; Q13, Q14, Q15, Q16

A QR17 is a real number. Finding its value employs sixteen real, non-integer, constants:

table five
C1 = 2 cos( 1 π/34) ≈ 1.991468  C9 = 2 cos( 9 π/34) ≈ 1.347391
C2 = 2 cos( 2 π/34) ≈ 1.965946 C10= 2 cos(10 π/34) ≈ 1.205269
C3 = 2 cos( 3 π/34) ≈ 1.923651 C11= 2 cos(11 π/34) ≈ 1.052864
C4 = 2 cos( 4 π/34) ≈ 1.864944 C12= 2 cos(12 π/34) ≈ 0.891477

C5 = 2 cos( 5 π/34) ≈ 1.790327 C13= 2 cos(13 π/34) ≈ 0.722483
C6 = 2 cos( 6 π/34) ≈ 1.700434 C14= 2 cos(14 π/34) ≈ 0.547326
C7 = 2 cos( 7 π/34) ≈ 1.596034 C15= 2 cos(15 π/34) ≈ 0.367499
C8 = 2 cos( 8 π/34) ≈ 1.478018 C16= 2 cos(16 π/34) ≈ 0.184537

The real-number value of a QR17 is then calculated as a linear combination:

Q = Q1C1 + Q2C2 + Q3C3 + … + Q16C16

In the table, the doubling of the cosine greatly simplifies the multiplication rule to come later.

(To reduce confusion, C should not be used as the name of a QR17.)

It is certainly true that if Q1 = R1, Q2 = R2, Q3 = R3, … Q16 = R16, then Q = R. However, the converse for QR17s is open question. Caution is required because for QR9s and QR15s (defined in the obvious manner), one of the cosines (namely cos π/3 = 12) is a rational number, and the converse is false. However, the converse is known to be valid for QR5s.

It so happens that, for any integer n,

n = ⟨ 0, +n, 0, −n; 0, +n, 0, −n; 0, +n, 0, −n; 0, +n, 0, −n

The sum and difference of two QR17s are:

Q + R = ⟨ Q1 + R1, Q2 + R2, Q3 + R3, … Q16 + R16
QR = ⟨ Q1R1, Q2R2, Q3R3, … Q16R16

An additive identity is 0 = ⟨ 0, 0, 0, … 0 ⟩.

The product of two QR17s is much more complicated than their sum or difference, even though it is derived merely by repeated application of elementary algebraic and trigonometric identities. The multiplication table for the Cn is therefore on a separate page due to its large size. Because QR17s are real numbers, multiplication distributes over addition, and this table is sufficient to define the product of any two QR17s.

This multiplicative identity is surprisingly complicated:

1 = ⟨ 0, +1, 0, −1; 0, +1, 0, −1; 0, +1, 0, −1; 0, +1, 0, −1 ⟩

Multiplication by an integer is straightforward:

nQ = ⟨ nQ1, nQ2, nQ3, … nQ16

Define two sets D1 and D2 such that:

 Q ∈ D1 if and only if Q is of the form ⟨ Q1, 0, Q3, 0; Q5, 0, Q7, 0; Q9, 0, Q11, 0; Q13, 0, Q15, 0 ⟩ Q ∈ D2 if and only if Q is of the form ⟨ 0, Q2, 0, Q4; 0, Q6, 0, Q8; 0, Q10, 0, Q12; 0, Q14, 0, Q16 ⟩

Not all QR17s belong to either of these sets, but those that do tell us something about the closure of addition, subtraction, and multiplication:

 If Q ∈ D1 and P ∈ D1, then Q + P ∈ D1 and Q − P ∈ D1. If Q ∈ D2 and P ∈ D2, then Q + P ∈ D2 and Q − P ∈ D2. If Q ∈ D1 and P ∈ D1, then QP ∈ D2. If Q ∈ D2 and P ∈ D2, then QP ∈ D2. If Q ∈ D2 and P ∈ D1, then QP ∈ D1. If Q ∈ D1 and P ∈ D1, then QP ∈ D1.

D2 establishes an integral domain in its own right.

Every QR17 can be expressed in radicals, and thus is algebraic. Hence a transcendental number like π can never be written as a QR17.

A number of identities can be more conveniently notated by first defining several ordered quadruples of real numbers:

• E1 = (C1, C5, C9, C13)
• E2 = (C2, C6, C10, C14)
• E3 = (C3, C7, C11, C15)
• E4 = (C4, C8, C12, C16)

The elements of a quadruple can be respectively raised to a power to form a new quadruple, for instance:

• E13 = (C13, C53, C93, C133)

The sum or product of the elements of a quadruple is written as in these examples:

• Σ E25 = C25 + C65 + C105 + C145
• Π E3−1 = C3−1 · C7−1 · C11−1 · C15−1

Each En is defined as a quadruple and not a set because there will be duplicate elements in the case of an En0.

Here are some identities involving integers:

table six
 Σ E2−8 + Σ E4−8 = 743724 Σ E2−6 + Σ E4−6 = 25362 Σ E2−4 + Σ E4−4 = 876 Σ E2−2 + Σ E4−2 = 36 Σ E20 + Σ E40 = 8 Σ E22 + Σ E42 = 15 Σ E24 + Σ E44 = 43 Σ E26 + Σ E46 = 138 Σ E28 + Σ E48 = 467
 Σ E2−7 − Σ E4−7 = −137155 Σ E2−5 − Σ E4−5 = −4654 Σ E2−3 − Σ E4−3 = −154 Σ E2−1 − Σ E4−1 = −4 Σ E21 − Σ E41 = 1 Σ E23 − Σ E43 = 4 Σ E25 − Σ E45 = 16 Σ E27 − Σ E47 = 64
 Σ E1−8 + Σ E3−8 = 3020 Σ E1−6 + Σ E3−6 = 414 Σ E1−4 + Σ E3−4 = 60 Σ E1−2 + Σ E3−2 = 12 Σ E10 + Σ E30 = 8 Σ E12 + Σ E32 = 17 Σ E14 + Σ E34 = 51 Σ E16 + Σ E36 = 170 Σ E18 + Σ E38 = 595
Π E2 · Π E4 = 1   Π E12 · Π E32 = 17

Because the QR17s form an integral domain, division is not guaranteed to work. Division by zero always fails as expected; otherwise, the components of a quotient might fail to be integers. In this case the components will be still rational numbers, and this fact suggests a workaround. For nonzero R, there exists a positive integer n such that (nQ) / R exists, even if n (Q / R) does not. Here are a few example reciprocals:

table seven
 1 / C2 = + C4 − C6 + C12 − C14 1 / C4 = + C6 − C8 + C10 − C12 1 / C6 = − C2 + C8 + C12 + C16 1 / C8 = + C10 − C12 + C14 − C16 1 / C10 = − C2 + C4 + C8 − C14 1 / C12 = + C2 − C4 + C10 − C16 1 / C14 = + C4 − C6 + C8 + C16 1 / C16 = + C2 + C6 + C10 + C14
 17 / C1 = 8 × C1 − 7 × C3 + 6 × C5 − 5 × C7 + 4 × C9 − 3 × C11 + 2 × C13 − 1 × C15 17 / C3 = 8 × C3 − 7 × C9 + 6 × C15 + 5 × C13 − 4 × C7 + 3 × C1 − 2 × C5 + 1 × C11 17 / C5 = 8 × C5 − 7 × C15 − 6 × C9 + 5 × C1 − 4 × C11 − 3 × C13 + 2 × C3 − 1 × C7 17 / C7 = 8 × C7 + 7 × C13 − 6 × C1 + 5 × C15 + 4 × C5 − 3 × C9 − 2 × C11 + 1 × C3 17 / C9 = 8 × C9 + 7 × C7 − 6 × C11 − 5 × C5 + 4 × C13 + 3 × C3 − 2 × C15 − 1 × C1 17 / C11 = 8 × C11 + 7 × C1 + 6 × C13 − 5 × C9 − 4 × C3 − 3 × C15 + 2 × C7 + 1 × C5 17 / C13 = 8 × C13 + 7 × C5 + 6 × C3 + 5 × C11 − 4 × C15 − 3 × C7 − 2 × C1 − 1 × C9 17 / C15 = 8 × C15 + 7 × C11 + 6 × C7 + 5 × C3 + 4 × C1 + 3 × C5 + 2 × C9 + 1 × C13

Many of these identities have counterparts for QRn when n is an odd integer other than 17.