Formula and examples obtained from "How to Calculate alt/az: SAAO" at
http://www.saao.ac.za/public-info/sun-moon-stars/sun-index/how-to-calculate-altaz/

examples-

(a) Cape Town       Feb 15   10:30   1995
(b) Bloemfontein    May 20   13:35   1996
(c) Johannesburg   Sept 25   16:45   1997


(1) find Y, the year minus 1900:

(a) Y = 95
(b)     96
(c)     97

(2) find Z(J) from this table:
   Jan   J= 1   Z(J)=  -0.5*        Jul   J=  7   Z(J) = 180.5
   Feb      2          30.5*        Aug       8          211.5
   Mar      3          58.5         Sep       9          242.5
   Apr      4          89.5         Oct      10          272.5
   May      5         119.5         Nov      11          303.5
   Jun      6         150.5         Dec      12          333.5
   (* reduce by one for a leap year)

(a) Z(J) =  30.5
(b)        119.5
(c)        242.5

(3) find D the number of days from this formula:
   D = integer(365.25 x Y) + Z(J) + K + UT/24
   where K is the day of the month and UT is the universal time

(a) D = int(365.25 x 95) +  30.5 + 15 +  8.500/24  = 34743.854
(b)     int(365.25 x 96) + 119.5 + 20 + 11.583/24  = 35203.983
(c)     int(365.25 x 97) + 242.5 + 25 + 14.750/24  = 35697.115

(4) find T the fraction of a julian century from this formula:
      T = D/36525

(a) T = 0.9512349
(b)     0.9638325
(c)     0.9773337


(5) find L the mean longitude of the sun from this formula:
   L = 279.697 + 36000.769 x T

(a) L = 34524.885  => 324.885   (removing multiples of 360 degrees)
(b)     34978.408  =>  58.408
(c)     35464.462  => 184.462

(6) find M the mean anomaly of the sun from this formula:
      M = 358.476 + 35999.050 x T

(a) M = 34602.029 =>  42.029   (removing multiples of 360 degrees)
(b)     35055.530 => 135.530
(c)     35541.561 => 261.561

(7) find epsilon the obliquity from this formula:
   epsilon = 23.452 - 0.013 x T

(a) epsilon = 23.4396
(b)           23.4395
(c)           23.4393

(8) find lambda the ecliptic longitude of the sun from this formula:
   lambda = L + (1.919 - 0.005 x T) x sin(M) + 0.020 x sin(2M)

(a) lambda = 324.885 + 1.9142 x  0.6695 + 0.020 x  0.9946 = 326.186
(b)           58.408 + 1.9142 x  0.7005 + 0.020 x -0.9998 =  59.729
(c)          184.462 + 1.9141 x -0.9892 + 0.020 x  0.2903 = 182.574

(9) find alpha the right ascension of the sun from this formula:
   alpha = arctan (tan(lambda) x cos(epsilon))      in same quadrant as lambda
(a) alpha = 328.428
(b)          57.537
(c)         182.362

(10) find delta the declination of the sun from this formula:
   delta = arcsin (sin(lambda) x sin(epsilon))

(a) DEC = -12.789
(b)        20.093
(c)        -1.024


(11) to proceed you need to know LONG the east-longitude of your location:

		east-longitude       latitude

Windhoek        17.10                -22.57
Cape Town       18.37                -33.92
P.E.            25.67                -33.97
Bloemfontein    26.12                -29.20
Johannesburg    28.00                -26.25
Durban          30.93                -29.92

(12) find HA the hour angle of the sun from this formula:
      HA = L - alpha + 180 + 15 x UT + LONG

(a) HA = 324.885 - 328.428 + 180 + 15 x  8.500 + 18.37  = -37.673
(b)       58.408 -  57.537 + 180 + 15 x 11.583 + 26.12  =  20.736
(c)      184.462 - 182.362 + 180 + 15 x 14.750 + 28.00  =  71.350

(13) find the altitude of the center of the sun ALT from this formula:

ALT [degrees] =
    ARCSIN [ SIN(LAT) x SIN(DEC)  +  COS(LAT) x COS(DEC) x COS(HA) ]

(a) ALT = ARCSIN ( -.5580 x -.2214  +  .8298 x .9752 x .7915 ) = 49.822
(b) ALT = ARCSIN ( -.4879 x  .3435  +  .8729 x .9391 x .9352 ) = 36.800
(c) ALT = ARCSIN ( -.4423 x -.0182  +  .8969 x .9998 x .3198 ) = 17.147


(14) find the azimuth of the sun AZ from this formula:

AZ [degrees] = ARCTAN [ SIN(HA) /
	      (COS(HA) x SIN(LAT)  -  TAN(DEC) x COS(LAT)) ]

(a) AZ = ARCTAN [ -.6112/ ( .7915 x -.5580  -  -.2270 x .8298 ) ]
       = ARCTAN (  2.4130 ) =  67.49   {i.e. east of true north}
(b) AZ = ARCTAN [  .3541/ ( .9352 x -.4879  -   .3658 x .8729 ) ]
       = ARCTAN ( -0.45656 ) = -24.54 = 335.46  {i.e. west of true north}
(c) AZ = ARCTAN [  .9475/ ( .3198 x -.4423  -  -.01787 x .8969 ) ]
       = ARCTAN ( -7.5546 ) = -82.46 = 277.54   {i.e. west of true north}

COMPARISON WITH COMPUTER ALMANAC PROGRAM

rough calculation here      computer almanac calculation    difference

(a) alt = 49.8, az =  67.5       ALT = 49.8, AZ =  67.5      none
(b) alt = 36.8, az = 335.5       ALT = 36.8, AZ = 335.5      none
(c) alt = 17.1, az = 277.5       ALT = 17.1, AZ = 277.5      none