Trigonometry

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Definitions of Trigonometric Functions 

By James D. Thomas

Draw a unit circle with center O. Let a central angle with initial side OP and terminal side OQ contain x radians (that is, the arc PQ has length x). Drop a perpendicular from Q to OP meeting it at R. Then OR = cos(x) and RQ = sin(x). If those directed line segments are up or to the right, the lengths are positive. If they are down or to the left, the lengths are negative.

Values at special angles:

  x      sin(x)     cos(x)      tan(x)      cot(x)       sec(x)       csc(x)
   
  0        0          1           0          ---           1           ---
 /6      1/2      sqrt(3)/2   sqrt(3)/3   sqrt(3)     2 sqrt(3)/3      2
 /4   sqrt(2)/2   sqrt(2)/2      1           1         sqrt(2)      sqrt(2)
 /3   sqrt(3)/2     1/2       sqrt(3)     sqrt(3)/3       2       2 sqrt(3)/3
 /2       1          0          ---          0           ---           1
2/3   sqrt(3)/2    -1/2      -sqrt(3)    -sqrt(3)/3      -2       2 sqrt(3)/3
3/4   sqrt(2)/2  -sqrt(2)/2     -1          -1        -sqrt(2)      sqrt(2)
5/6      1/2     -sqrt(3)/2  -sqrt(3)/3  -sqrt(3)    -2 sqrt(3)/3      2
          0         -1           0          ---          -1           ---
More values at special angles:
  x              /10                  /5
sin(x)      (-1+sqrt[5])/4      sqrt(10-2 sqrt[5])/4
cos(x)   sqrt(10+2 sqrt[5])/4      (1+sqrt[5])/4
tan(x)    sqrt(1-2/sqrt[5])      sqrt(5-2 sqrt[5])
cot(x)    sqrt(5+2 sqrt[5])      sqrt(1+2/sqrt[5])
sec(x)    sqrt(2-2/sqrt[5])         -1+sqrt[5]
csc(x)        1+sqrt[5]          sqrt(2+2/sqrt[5])
Use the above values and the identities below to obtain values of trigonometric functions of the following multiples of /10:

  
3/10 = /2 - /5,
2/5  = /2 - /10,
3/5  = /2 + /10,
7/10 = /2 + /5,
4/5  =    - /5,
9/10 =    - /10.

Bounds

   |sin(x)| <= 1,
   |cos(x)| <= 1,
   |sec(x)| >= 1,
   |csc(x)| >= 1.

Identities

   sec(x) = 1/cos(x),
   csc(x) = 1/sin(x),
   cot(x) = 1/tan(x),
   tan(x) = sin(x)/cos(x),
   cot(x) = cos(x)/sin(x).

   sin(-x) = -sin(x),
   cos(-x) = cos(x),
   tan(-x) = -tan(x),
   cot(-x) = -cot(x),
   sec(-x) = sec(x),
   csc(-x) = -csc(x).

   sin(/2-x) = cos(x),
   cos(/2-x) = sin(x),
   tan(/2-x) = cot(x),
   cot(/2-x) = tan(x),
   sec(/2-x) = csc(x),
   csc(/2-x) = sec(x).

   sin(/2+x) = cos(x),
   cos(/2+x) = -sin(x),
   tan(/2+x) = -cot(x),
   cot(/2+x) = -tan(x),
   sec(/2+x) = -csc(x),
   csc(/2+x) = sec(x).

   sin(-x) = sin(x),
   cos(-x) = -cos(x),
   tan(-x) = -tan(x),
   cot(-x) = -cot(x),
   sec(-x) = -sec(x),
   csc(-x) = csc(x).

   sin(+x) = -sin(x),
   cos(+x) = -cos(x),
   tan(+x) = tan(x),
   cot(+x) = cot(x),
   sec(+x) = -sec(x),
   csc(+x) = -csc(x).

   sin(2+x) = sin(x),
   cos(2+x) = cos(x),
   tan(2+x) = tan(x),
   cot(2+x) = cot(x),
   sec(2+x) = sec(x),
   csc(2+x) = csc(x).

   sin2(x) + cos2(x) = 1,
   tan2(x) + 1 = sec2(x),
   1 + cot2(x) = csc2(x).

   sin(x+y) = sin(x)cos(y) + cos(x)sin(y),
   cos(x+y) = cos(x)cos(y) - sin(x)sin(y),
   tan(x+y) = [tan(x)+tan(y)]/[1-tan(x)tan(y)],
   cot(x+y) = [cot(x)cot(y)-1]/[cot(x)+cot(y)].

   sin(x-y) = sin(x)cos(y) - cos(x)sin(y),
   cos(x-y) = cos(x)cos(y) + sin(x)sin(y),
   tan(x-y) = [tan(x)-tan(y)]/[1+tan(x)tan(y)],
   cot(x-y) = [cot(x)cot(y)+1]/[cot(y)-cot(x)].

   sin(2x) = 2 sin(x)cos(x),
   cos(2x) = cos2(x) - sin2(x),
            = 2 cos2(x) - 1,
            = 1 - 2 sin2(x),
   tan(2x) = [2 tan(x)]/[1-tan2(x)],
   cot(2x) = [cot2(x)-1]/[2 cot(x)].


   |sin(x/2)| = sqrt([1-cos(x)]/2),
   
   |cos(x/2)| = sqrt([1+cos(x)]/2),
   
   |tan(x/2)| = sqrt([1-cos(x)]/[1+cos(x)]),
   
   tan(x/2) = [1-cos(x)]/sin(x),
            = sin(x)/[1+cos(x)].
            

   sin(3x) = 3 sin(x) - 4 sin3(x),
   cos(3x) = 4 cos3(x) - 3 cos(x),
   tan(3x) = [3 tan(x)-tan3(x)]/[1-3 tan2(x)].

   sin(4x) = 4 sin(x)cos(x)[2 cos2(x)-1],
   cos(4x) = 8 cos4(x) - 8 cos2(x) + 1.

   sin(5x) = 5 sin(x) - 20 sin3(x) + 16 sin5(x),
   cos(5x) = 16 cos5(x) - 20 cos3(x) + 5 cos(x).

   sin(6x) = 2 sin(x)cos(x)[16 cos4(x) - 16 cos2(x) + 3],
   cos(6x) = 32 cos6(x) - 48 cos4(x) + 18 cos2(x) - 1.

   sin(nx) = 2 sin([n-1]x)cos(x) - sin([n-2]x),
   cos(nx) = 2 cos([n-1]x)cos(x) - cos([n-2]x),
   tan(nx) = (tan[(n-1)x]+tan[x])/(1-tan[(n-1)x]tan[x]).

   sin(x)cos(y) = [sin(x+y) + sin(x-y)]/2,
   cos(x)sin(y) = [sin(x+y) - sin(x-y)]/2,
   cos(x)cos(y) = [cos(x-y) + cos(x+y)]/2,
   sin(x)sin(y) = [cos(x-y) - cos(x+y)]/2.

   sin(x) + sin(y) = 2 sin[(x+y)/2]cos[(x-y)/2],
   sin(x) - sin(y) = 2 cos[(x+y)/2]sin[(x-y)/2],
   cos(x) + cos(y) = 2 cos[(x+y)/2]cos[(x-y)/2],
   cos(x) - cos(y) = -2 sin[(x+y)/2]sin[(x-y)/2],
   tan(x) + tan(y) = sin(x+y)/[cos(x)cos(y)],
   tan(x) - tan(y) = sin(x-y)/[cos(x)cos(y)],
   cot(x) + cot(y) = sin(x+y)/[sin(x)sin(y)],
   cot(x) - cot(y) = -sin(x-y)/[sin(x)sin(y)].

   [sin(x)+sin(y)]/[cos(x)+cos(y)] = tan[(x+y)/2],
   [sin(x)-sin(y)]/[cos(x)+cos(y)] = tan[(x-y)/2],
   [sin(x)+sin(y)]/[cos(x)-cos(y)] = -cot[(x-y)/2],
   [sin(x)-sin(y)]/[cos(x)-cos(y)] = -cot[(x+y)/2],
   [sin(x)+sin(y)]/[sin(x)-sin(y)] = tan[(x+y)/2]/tan[(x-y)/2].

   sin2(x) - sin2(y) = sin(x+y)sin(x-y),
   cos2(x) - cos2(y) = -sin(x+y)sin(x-y),
   cos2(x) - sin2(y) = cos(x+y)cos(x-y).
   
   sin2(x) = (1 - cos[2x])/2,
   cos2(x) = (1 + cos[2x])/2,
   tan2(x) = (1 - cos[2x])/(1 + cos[2x]),

   sin3(x) = (3 sin[x] - sin[3x])/4,
   cos3(x) = (3 cos[x] + cos[3x])/4,

   sin4(x) = (3 - 4 cos[2x] + cos[4x])/8,
   cos4(x) = (3 + 4 cos[2x] + cos[4x])/8,

   sin5(x) = (10 sin[x] - 5 sin[3x] + sin[5x])/16,
   cos5(x) = (10 cos[x] + 5 cos[3x] + cos[5x])/16,

   sin6(x) = (10 - 15 cos[2x] + 6 cos[4x] - cos[6x])/32,
   cos6(x) = (10 + 15 cos[2x] + 6 cos[4x] + cos[6x])/32,

Relations in Right Triangles

In the right triangle ABC with right angle C = /2, 

   A + B = /2,
   c2 = a2 + b2,
   sin(A) = cos(B) = a/c,
   cos(A) = sin(B) = b/c,
   tan(A) = cot(B) = a/b,
   cot(A) = tan(B) = b/a,
   sec(A) = csc(B) = c/b,
   csc(A) = sec(B) = c/a,
   ha = b,
   hb = a,
   hc = ab/c.

Solving Right Triangles
Case I: You are given a and A.
B = /2 - A, c = a csc(A), b = a cot(A).

 

Case II: You are given a and B.
A = /2 - B, c = a sec(B), b = a tan(B).

 

Case III: You are given c and A.
B = /2 - A, a = c sin(A), b = c cos(A).

 

Case IV: You are given a and b.
tan(A) = a/b, B = /2 - A, c = a csc(A).

 

Case V: You are given a and c.
sin(A) = a/c, B = /2 - A, b = a cot(A).

Relations in Oblique Triangles

   A + B + C = ,
   s = (a+b+c)/2, half the perimeter,
   r = radius of inscribed circle,
   R = radius of circumscribed circle,
   K = area.
The Law of Sines:
   a/sin(A) = b/sin(B) = c/sin(C) = 2R.
This implies that a <= b <= c if and only if A <= B <= C.

The Law of Cosines:

   a2 = b2 + c2 - 2bc cos(A),
   b2 = c2 + a2 - 2ca cos(B),
   c2 = a2 + b2 - 2ab cos(C).
The Law of Tangents:
   (a+b)/(a-b) = tan[(A+B)/2]/tan[(A-B)/2],
   (b+c)/(b-c) = tan[(B+C)/2]/tan[(B-C)/2],
   (c+a)/(c-a) = tan[(C+A)/2]/tan[(C-A)/2].
Newton's Formulae:
   (a+b)/c = cos[(A-B)/2]/sin(C/2),
   (b+c)/a = cos[(B-C)/2]/sin(A/2),
   (c+a)/b = cos[(C-A)/2]/sin(B/2).
Mollweide's Equations:
   (a-b)/c = sin[(A-B)/2]/cos(C/2),
   (b-c)/a = sin[(B-C)/2]/cos(A/2),
   (c-a)/b = sin[(C-A)/2]/cos(B/2).
Other relations:
   a = b cos(C) + c cos(B),
   b = c cos(A) + a cos(C),
   c = a cos(B) + b cos(A).

   tan[(A-B)/2] = [(a-b)/(a+b)]cot(C/2),
   tan[(B-C)/2] = [(b-c)/(b+c)]cot(A/2),
   tan[(C-A)/2] = [(c-a)/(c+a)]cot(B/2).

   sin(A) = 2K/(bc),
   sin(B) = 2K/(ca),
   sin(C) = 2K/(ab).

   K = sr = sqrt[s(s-a)(s-b)(s-c)],
   K = aha/2 = bhb/2 = chc/2,
   K = ab sin(C)/2 = bc sin(A)/2 = ca sin(B)/2,
   K = a2 sin(B)sin(C)/[2 sin(A)],
     = b2 sin(C)sin(A)/[2 sin(B)],
     = c2 sin(A)sin(B)/[2 sin(C)].

   R = abc/(4K) = a/[2 sin(A)] = b/[2 sin(B)] = c/[2 sin(C)],
   r = K/s,
     = sqrt[(s-a)(s-b)(s-c)/s],
     = c sin(A/2)sin(B/2)/cos(C/2),
     = ab sin(C)/(2s),
     = (s-c)tan(C/2).

   sin(A/2) = sqrt[(s-b)(s-c)/(bc)],
   sin(B/2) = sqrt[(s-c)(s-a)/(ca)],
   sin(C/2) = sqrt[(s-a)(s-b)/(ab)].

   cos(A/2) = sqrt[s(s-a)/(bc)],
   cos(B/2) = sqrt[s(s-b)/(ca)],
   cos(C/2) = sqrt[s(s-c)/(ab)].

   tan(A/2) = sqrt[(s-b)(s-c)/{s(s-a)}] = r/(s-a),
   tan(B/2) = sqrt[(s-c)(s-a)/{s(s-b)}] = r/(s-b),
   tan(C/2) = sqrt[(s-a)(s-b)/{s(s-c)}] = r/(s-c).

   (a+b)/(a-b) = [sin(A)+sin(B)]/[sin(A)-sin(B)] = cot(C/2)/tan[(A-B)/2],
   (b+c)/(b-c) = [sin(B)+sin(C)]/[sin(B)-sin(C)] = cot(A/2)/tan[(B-C)/2],
   (c+a)/(c-a) = [sin(C)+sin(A)]/[sin(C)-sin(A)] = cot(B/2)/tan[(C-A)/2].

   ha = a sin(B)sin(C)/sin(B+C) = a/[cot(B)+cot(C)] = b sin(C) = c sin(B),
   hb = b sin(C)sin(A)/sin(C+A) = b/[cot(C)+cot(A)] = c sin(A) = a sin(C),
   hc = c sin(A)sin(B)/sin(A+B) = c/[cot(A)+cot(B)] = a sin(B) = b sin(A).
   
   cos(A) + cos(B) + cos(C) = 1 + r/R.

Solving Oblique Triangles

Case I: You are given any two angles and one side c.

 

The third angle is determined from A + B + C = . Now the Law of Sines can be used to find b = c sin(B)/sin(C) and a = c sin(A)/sin(C).

 

Case II: You are given two sides and the angle opposite one of them, say a, c, and A.

 

Subcase A: a < c sin(A). There is no solution.

 

Subcase B: a = c sin(A). There is one solution:
C = /2, B = /2 - A, b = c cos(A).

 

Subcase C: c > a > c sin(A). There are two solutions. Use the Law of Sines to find sin(C) = c sin(A)/a < 1. There are two angles C and C' = - C having that sine, one acute and one obtuse. Then compute B = - A - C and B' = - A - C'. Now use the Law of Sines again to find b = a sin(B)/sin(A) and b' = a sin(B')/sin(A). The solutions are (a,b,c,A,B,C) and (a,b',c,A,B',C').

 

Subcase D: a >= c. There is one solution. Use the Law of Sines to find sin(C) = c sin(A)/a <= 1. Then angle C must be acute, so it can be found uniquely from sin(C). Then compute B = - A - C. Now use the Law of Sines again to find b = a sin(B)/sin(A).

 

Case III: You are given two sides and the included angle, say a, b, and C.

 

You can compute the third side c by using the Law of Cosines. Then the Law of Sines can be used to find the sines of the other two angles sin(A) = a sin(C)/c and sin(B) = b sin(C)/c. The angles opposite the two shortest sides are then acute, and uniquely determined from their sines, and the third, largest angle is found from A + B + C = .

 

Alternatively, you can use the Law of Tangents. You know that (A+B)/2 = (-C)/2, which is easily computable. Then by the Law of Tangents, tan[(A-B)/2] = cot(C/2) (a-b)/(a+b), so you can find (A-B)/2 uniquely. Then A = (-C)/2 + (A-B)/2, and B = (-C)/2 - (A-B)/2. Then c = a sin(C)/sin(A).

 

Case IV: You are given all three sides.

 

You can use the Law of Cosines to find A, then use the Law of Sines to compute sin(B) = b sin(A)/a and sin(C) = c sin(A)/a.

 

Alternatively, you can find r = sqrt[(s-a)(s-b)(s-c)/s], and use tan(A/2) = r/(s-a) to find A/2, and hence A, and similarly for B and C.

 

Alternatively, you can use sin(A/2) = sqrt[(s-b)(s-c)/(bc)] to find A/2 (since A/2 < /2), and hence A, and similarly for B and C.

In any case, the results can be checked by using Mollweide's Equations.

Inverse Trigonometric Functions

   x = Arcsin(y)  ==>  y = sin(x),  -/2 <= x <= /2,
   x = Arccos(y)  ==>  y = cos(x),     0 <= x <= ,
   x = Arctan(y)  ==>  y = tan(x),  -/2 <  x <  /2,
   x = Arccot(y)  ==>  y = cot(x),     0 <  x <  ,
   x = Arcsec(y)  ==>  y = sec(x),     0 <  x <  ,
   x = Arccsc(y)  ==>  y = csc(x),  -/2 <  x <  /2.

   y = sin(x)  ==>  x = Arcsin(y) + 2n  or   - Arcsin(y) + 2n,
   y = cos(x)  ==>  x = Arccos(y) + 2n  or  -Arccos(y)    + 2n,
   y = tan(x)  ==>  x = Arctan(y) +  n,
   y = cot(x)  ==>  x = Arccot(y) +  n,
   y = sec(x)  ==>  x = Arcsec(y) + 2n  or  -Arcsec(y)    + 2n,
   y = csc(x)  ==>  x = Arccsc(y) + 2n  or   - Arccsc(y) + 2n,
where n is an arbitrary integer.
   Arcsin(y) + Arccos(y) = /2,
   Arctan(y) + Arccot(y) = /2,
   Arcsec(y) + Arccsc(y) = /2.

   Arcsin(y) = Arccsc(1/y),
   Arccos(y) = Arcsec(1/y),
   Arctan(y) = Arccot(1/y),  y > 0
   Arccot(y) = Arctan(1/y),  y > 0
   Arcsec(y) = Arccos(1/y),
   Arccsc(y) = Arcsin(1/y).

   Arcsin[sin(x)] = x - Floor[x/ + 1/2],
   Arccos[cos(x)] = x - Floor[x/],
   Arctan[tan(x)] = x - Floor[x/ + 1/2].

   sin[Arcsin(y)] = cos[Arccos(y)] = y,
   cos[Arcsin(y)] = sin[Arccos(y)] = sqrt[1-y2],
   tan[Arcsin(y)] = cot[Arccos(y)] = y/sqrt[1-y2],
   cot[Arcsin(y)] = tan[Arccos(y)] = sqrt[1-y2]/y,
   sec[Arcsin(y)] = csc[Arccos(y)] = 1/sqrt[1-y2],
   csc[Arcsin(y)] = sec[Arccos(y)] = 1/y,
   sin[Arctan(y)] = cos[Arccot(y)] = y/sqrt[1+y2],
   cos[Arctan(y)] = sin[Arccot(y)] = 1/sqrt[1+y2],
   tan[Arctan(y)] = cot[Arccot(y)] = y,
   cot[Arctan(y)] = tan[Arccot(y)] = 1/y,
   sec[Arctan(y)] = csc[Arccot(y)] = sqrt[1+y2],
   csc[Arctan(y)] = sec[Arccot(y)] = sqrt[1+y2]/y,
   sin[Arcsec(y)] = cos[Arccsc(y)] = sqrt[y2-1]/y,
   cos[Arcsec(y)] = sin[Arccsc(y)] = 1/y,
   tan[Arcsec(y)] = cot[Arccsc(y)] = sqrt[y2-1],
   cot[Arcsec(y)] = tan[Arccsc(y)] = 1/sqrt[y2-1],
   sec[Arcsec(y)] = csc[Arccsc(y)] = y,
   csc[Arcsec(y)] = sec[Arccsc(y)] = y/sqrt[y2-1].

   sin[2 Arcsin(y)] = 2y sqrt[1-y2],
   cos[2 Arccos(y)] = 2y2 - 1,
   tan[2 Arctan(y)] = 2y/(1-y2).
 

Trig Numerical Table


Degrees Radian Measure Sin Cos Tan   Degrees Radian Measure Sin Cos Tan
0 0.00000 0.00000 1.00000 0.00000   46 0.80285 0.71934 0.69466 1.03553
1 0.01745 0.01745 0.99985 0.01746   47 0.82030 0.73135 0.68200 1.07237
2 0.03491 0.03490 0.99939 0.03492   48 0.83776 0.74314 0.66913 1.11061
3 0.05236 0.05234 0.99863 0.05241   49 0.85521 0.75471 0.65606 1.15037
4 0.06981 0.06976 0.99756 0.06993   50 0.87266 0.76604 0.64279 1.19175
5 0.08727 0.08716 0.99619 0.08749   51 0.89012 0.77715 0.62932 1.23490
6 0.10472 0.10453 0.99452 0.10510   52 0.90757 0.78801 0.61566 1.27994
7 0.12217 0.12187 0.99255 0.12278   53 0.92502 0.79864 0.60182 1.32704
8 0.13963 0.13917 0.99027 0.14054   54 0.94248 0.80902 0.58779 1.37638
9 0.15708 0.15643 0.98769 0.15838   55 0.95993 0.81915 0.57358 1.42815
10 0.17453 0.17365 0.98481 0.17633   56 0.97738 0.82904 0.55919 1.48256
11 0.19199 0.19081 0.98163 0.19438   57 0.99484 0.83867 0.54464 1.53986
12 0.20944 0.20791 0.97815 0.21256   58 1.01229 0.84805 0.52992 1.60033
13 0.22689 0.22495 0.97437 0.23087   59 1.02974 0.85717 0.51504 1.66428
14 0.24435 0.24192 0.97030 0.24933   60 1.04720 0.86603 0.50000 1.73205
15 0.26180 0.25882 0.96593 0.26795   61 1.06465 0.87462 0.48481 1.80405
16 0.27925 0.27564 0.96126 0.28675   62 1.08210 0.88295 0.46947 1.88073
17 0.29671 0.29237 0.95630 0.30573   63 1.09956 0.89101 0.45399 1.96261
18 0.31416 0.30902 0.95106 0.32492   64 1.11701 0.89879 0.43837 2.05030
19 0.33161 0.32557 0.94552 0.34433   65 1.13446 0.90631 0.42262 2.14451
20 0.34907 0.34202 0.93969 0.36397   66 1.15192 0.91355 0.40674 2.24604
21 0.36652 0.35837 0.93358 0.38386   67 1.16937 0.92050 0.39073 2.35585
22 0.38397 0.37461 0.92718 0.40403   68 1.18682 0.92718 0.37461 2.47509
23 0.40143 0.39073 0.92050 0.42447   69 1.20428 0.93358 0.35837 2.60509
24 0.41888 0.40674 0.91355 0.44523   70 1.22173 0.93969 0.34202 2.74748
25 0.43633 0.42262 0.90631 0.46631   71 1.23918 0.94552 0.32557 2.90421
26 0.45379 0.43837 0.89879 0.48773   72 1.25664 0.95106 0.30902 3.07768
27 0.47124 0.45399 0.89101 0.50953   73 1.27409 0.95630 0.29237 3.27085
28 0.48869 0.46947 0.88295 0.53171   74 1.29154 0.96126 0.27564 3.48741
29 0.50615 0.48481 0.87462 0.55431   75 1.30900 0.96593 0.25882 3.73205
30 0.52360 0.50000 0.86603 0.57735   76 1.32645 0.97030 0.24192 4.01078
31 0.54105 0.51504 0.85717 0.60086   77 1.34390 0.97437 0.22495 4.33148
32 0.55851 0.52992 0.84805 0.62487   78 1.36136 0.97815 0.20791 4.70463
33 0.57596 0.54464 0.83867 0.64941   79 1.37881 0.98163 0.19081 5.14455
34 0.59341 0.55919 0.82904 0.67451   80 1.39626 0.98481 0.17365 5.67128
35 0.61087 0.57358 0.81915 0.70021   81 1.41372 0.98769 0.15643 6.31375
36 0.62832 0.58779 0.80902 0.72654   82 1.43117 0.99027 0.13917 7.11537
37 0.64577 0.60182 0.79864 0.75355   83 1.44862 0.99255 0.12187 8.14435
38 0.66323 0.61566 0.78801 0.78129   84 1.46608 0.99452 0.10453 9.51436
39 0.68068 0.62932 0.77715 0.80978   85 1.48353 0.99619 0.08716 11.43005
40 0.69813 0.64279 0.76604 0.83910   86 1.50098 0.99756 0.06976 14.30067
41 0.71558 0.65606 0.75471 0.86929   87 1.51844 0.99863 0.05234 19.08114
42 0.73304 0.66913 0.74314 0.90040   88 1.53589 0.99939 0.03490 28.63625
43 0.75049 0.68200 0.73135 0.93252   89 1.55334 0.99985 0.01745 57.28996
44 0.76794 0.69466 0.71934 0.96569   90 1.57080 1.00000 0.00000  
45 0.78540 0.70711 0.70711 1.00000            

 

 

 

 



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144B W. Walnut Fullerton, Calif. 92634-6280
Tel: 714-525-0608
FAX: 714-525-0628

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Last modified: April 11, 2003