arctan

[f.sub.ap(e,o)4] = [2[f.sub.a0]/[pi]] [[pi] - arctan [square root of (-[B.sub.(e,o)] + [square root of ([B.sup.2.sub.(e,o)] - 4[A.sub.(e,o)][C.sub.(e,o)])]/2[A.sub.(e,o)])]] (4d)

[[alpha].sub.2] = arctan [h/[z + l cos [theta]]] [right arrow] [[alpha].sub.2] [approximately equal to] [h/[z + l cos [theta]]] (4)

So [phi] = [gamma]/2[pi] (arctan x + b/z - arctan x - b/z) (4)

+ arctan = (BR + [k.sub.d][C.sub.e])/JR[[omega].sub.c] = - [pi]/2 + [[phi].sub.m], (17)

[theta]' = arctan (r cos [[theta].sub.n,m] sin [[phi].sub.n,m]/ r cos [[theta].sub.n,m] cos [[phi].sub.n,m] + (n- 1)[DELTA]), (16)

[[theta]'.sub.1] = arctan (y/x) [+ or -] [beta], (5)

Physically, this ring is interpreted as the superposition of all the plane waves in the McCutchen sphere whose wavevectors of modulus k lie on a conical surface of angle [[theta].sub.0] = arctan ([k.sub.t]/[k.sub.z]) with respect to z axis and satisfying [k.sup.2] = [k.sup.2.sub.z] + [k.sup.2.sub.t] [29].

[[beta].sub.B, i] = arctan ([R.sub.cell]cos[[theta].sub.c, i]/[H.sub.HAPS]) for i = 2, ...

Z (u) [equivalent to] {[tanh.sup.-1] [square root of (-u)]/[(-3c).sup.1/4] - arctan [square root of (-u)]/[(- 3c).sup.1/4]) = [(-3c).sup.1/4]/2[square root of (3[alpha])] [absolute value of ([xi])].

[F.sub.[summation over] erdvine padeti nusako kampas [[alpha].sub.s] = arctan ([F.sub.B]/[F.sub.A]).

Probabilities that random intersections of a cube have a given number of vertices Number of Intersection probability polygon found derived theoretically vertices numerically 3 0.2798 2 - 4[square root of (2)]/[pi] arctan [square root of (2)] [approximately equal to] 0.2798 4 0.4873 2/[square root of (3)]-3-2[square root of (2)] + 12[square root of (2)]/[pi] arctan [square root of (2)] [approximately equal to] 0.4868 5 0.1865 2 - 4/[square root of (3)] + 4[square root of (2)] - 12[square root of (2)]/[pi] arctan [square root of (2)] [approximately equal to] 0.01869 6 0.0464 2/[square root of (3)] - 2[square root of (2)] + 4[square root of (2)]/[pi] arctan [square root of (2)] [approximately equal to] 0.0464 Table 4.

NIntegrate Bound error arctan x -0.136146 -0.136146 -2.75111 x [10.sup.-10]

To illustrate, for each point (x, y, z) define the angle [theta] = arctan x/z [member of] (-[pi], [pi]] with the signs of x and z used to determine the appropriate quadrant for the angle.

[psi] = arctan [alpha] x sin [THETA] / 1 + [alpha] x cos [THETA],

The maximum polymer length for no fiber fracture is 2L, and when we allow for the C-C bond angle of (109.5 [degrees]) or 2 arctan [-square root of 2] the maximum number of carbon atoms per chain, [N.sub.max] is 2[-square root of 3] L/[a.sub.o] where [a.sub.o] is the length of the C-C single bond.