Considering these ideas of navigation.. What is your opnion on it?

[Paul Elliott]

Local GPS is occasionally degraded or disabled when the U.S. Navy is having large exercises around the Hawaiian Islands. They announce this well beforehand.

I also heard that some of the Iridium constellation had orbits adjusted for better coverage in the middle east during the Gulf War(s).

[14murs14]

Originally Posted by Paul Elliott

Local GPS is occasionally degraded or disabled when the U.S. Navy is having large exercises around the Hawaiian Islands. They announce this well beforehand.

I also heard that some of the Iridium constellation had orbits adjusted for better coverage in the middle east during the Gulf War(s).

I sincerely doubt that happens anymore, I flew out of Hawaii for the Coast Guard from 2003-2006 and GPS was approved for en-route guidance in IMC, which couldn't have been the case if they were degrading or disabling it in any way. Don't recall ever seeing a NOTAM about that, but could have happened when I was on leave I guess?

Iridium is a private constellation owned by a private company, unlike the GPS constellation which is owned by the U.S. government. They're also in a low earth orbit at 485 miles vice 12,000 miles for GPS satellites and as a result have a constellation of 70+ from initial operation vice the 19 GPS satellites around during Desert Storm. And they didn't go into operation until 1998 well after Desert Storm. I guess if DoD paid them enough they'd adjust orbits for better coverage in the more recent never-ending wars, but with a LEO constellation like that it really wouldn't make much sense from an orbital mechanics standpoint, especially since they rely on inter-satellite links which would be screwed up by changing the constellation orientation. Would be interested to see any documentation on this?

[Paul Elliott]

Originally Posted by 14murs14

I sincerely doubt that happens anymore, I flew out of Hawaii for the Coast Guard from 2003-2006 and GPS was approved for en-route guidance in IMC, which couldn't have been the case if they were degrading or disabling it in any way. Don't recall ever seeing a NOTAM about that, but could have happened when I was on leave I guess?

It was announced during the 2008 and I believe 2010 RIMPAC exercises (Rim of the Pacific Exercise, is the world's largest international maritime warfare exercise and takes place in even-numbered years), and perhaps others. I was there in 2008, 2010, 2012, and 2014 for the Pacific Cup sailboat race. They gave the areas which might be affected, but I don't know if the degradation actually occurred. We were definitely warned.

[hellosailor]

"The US would turn off their GPS network in a heartbeat if an enemy had the potential to use it for GPS guided attack."
Not would, but could.

There were supposedly over 200 incidents of GPS jamming AND SPOOFING in the US alone last year. Jamming and spoofing are easy.

Similarly, if I learned you were sending a GPS-guided drone my way? No need to degrade the signal or shut the system off. Shutting the system off might simply cause the drone go switch to a backup navigation system, and still hit the target.

Far more reliable to simply corrupt the GPS stream, and tell the satellites "The earth just shifted two miles up and over" and let the evil drone adjust itself to target the wrong place. If that can't be done with one phone call, as of right now, then someone needs to be demoted about five grades in rank.

[Paul Elliott]

Originally Posted by 14murs14

Iridium is a private constellation owned by a private company, unlike the GPS constellation which is owned by the U.S. government. They're also in a low earth orbit at 485 miles vice 12,000 miles for GPS satellites and as a result have a constellation of 70+ from initial operation vice the 19 GPS satellites around during Desert Storm. And they didn't go into operation until 1998 well after Desert Storm. I guess if DoD paid them enough they'd adjust orbits for better coverage in the more recent never-ending wars, but with a LEO constellation like that it really wouldn't make much sense from an orbital mechanics standpoint, especially since they rely on inter-satellite links which would be screwed up by changing the constellation orientation. Would be interested to see any documentation on this?

I can't find any references to Iridium orbit changes, and I can't recall where I thought I heard that. Perhaps I misunderstood. And your point about the orbital mechanics does make sense.

[Cadence]

Originally Posted by SteveG58

Using a plumb bob on a board marked out with a protractor can give you a fairly accurate reading. Midday only I think!

Probably on flat seas and you have Erik the Red aboard.

[GrowleyMonster]

FIRST, learn to keep a proper DR on a chart or plot sheet. This is the cornerstone of primitive navigation and should also be the cornerstone of electronic navigation, too, but sadly nobody practices keeping a DR anymore.



Without a DR, you cannot have a running fix. A running fix is two or more LOP, Lines Of Position, advanced or retarded to a single time so that they point where they cross is the celestial running fix for that particular time. From the new fix, you can compare to the DR position for that time, and calculate set and drift and leeway.



The simplest way to use celestial navigation is to find your Latitude at LAN, Local Apparent Noon. The cool thing about it is you do not even have to know what time LAN is at your position or indeed even know your position. Before chronometers, the average navigator made do with nothing more than a DR track and daily Latitude checks. There were ways of determining time and therefore Longitude, using precomputed eclipses of various celestial bodies, and also by Lunar Distance calculation which was not a simple endeavor. A crude method of implementing the difference in changing LHA, Local Hour Angle, between the moon and another body, has already been mentioned, and it can easily get you within well under a minute of the correct time and therefore about 15 minutes of Longitude, but even with a calculator you go through several iterations to nail it down with that much precision.


Also previously posted are methods of determining accurate time by recording error and rate of a watch, and keeping more than one. And your radio can give you a very good time reference, in the time ticks of several stations that can be picked up worldwide at various times. BBC is a very good example of a time check apart from WWV, WWVH, and other stations maintained by various governments. Be listening at the top of the hour and you have a great time check.


While you do still have use of the GPS, you can use the time by GPS, which is not as accurate as the radio check, but will still be within a second of correct, an incosequential error compared to the expected error of your sextant observations from a small boat.


Along with DR and celestial, you should also familiarize yourself with other techniques such as use of a sounding lead, and radio bearings. Observations of the world around you such as prevailing winds, planes flying overhead, ships sighted, flotsam, wave action, birds, fish and marine mammals seen, all can give clues to your position. Learn to read a pilot chart and you can closely estimate current and also make a good general guess at position by the prevailing wind.



But back to celestial. Once you have time, you can advance a morning Sun line to local Noon, advance or retard the LAN latitude line, and retard an afternoon Sun line for your noon fix. So with only observations of the Sun, you can keep a fairly good position. But even more, you can take a round of sights at morning and evening twilight of stars, planets, and/or moon for an evening or morning fix. Latitude by Polaris is fairly straightforward, as well.


It is good to be able to compute azimuth and intercept for an LOP using a calculator and two formulae. With practice you can reduce a sight quickly. You can even mathematically calculate a fix position without use of chart or plotting tools, though it is somewhat involved. But in the spirit of going primitive, without electronic age aids, you also need to be able to use sight reduction tables to solve your sights. Most popular is HO229, a multivolume set. It is nothing but a tremendous list of precalculated solutions so it never goes out of date. What does go out of date is the Nautical Almanac. It is still usble with limitations even if it is out of date, but the current one will be much more useful. Again, electronics can aid you... you can print out the almanac pages you might need for your voyage ahead of time.



Okay so now you got sextant, sight reduction tables, calculator, (which you can do without if necessary) and almanac data either printed yourself or in the current year's almanac. You also need plotting tools. At a minimum, a pair of triangles and a set of dividers. And a chart, or a plot sheet which you can construct yourself or buy.



For accuracy, you should also have weather observation tools such as barometer and thermometer (wet and dry bulb).



None of this gear is worth nickel if you don't practice its use. A standard worksheet helps, but you still need the familiarity that only repetitious practice can give.



A Davis sextant? Well, it is functional, and better than making one out of a pencil and protractor LOL. Sailors have indeed navigated across oceans with this modest plastic sextant. However, for well under a grand you can have a brand new Astra, which is a very good instrument. Or a used Plath, Freiberger, Tamaya, or other professional grade sextant with calibration. Okay, yeah, except under ideal conditions an observation from a small boat won't be all that accurate anyway. Nevertheless, a few hundred bucks for a more accurate instrument is IMHO money well spent if you are serious about celestial navigation.


A fancy chronometer is the one area where you CAN save money. The average digital watch is more accurate, and even the rate is more consistent. The rock steady rate of a chronometer used to be its big selling point even when a Timex windup was more accurate on a daily basis. Modern digital watches have basically unchanging rates even when the battery is nearly dead. So if you have say a rate of plus 15 seconds a month you know it will be another 15 seconds the next month and another 15 seconds the next... and so, you calculate the chronometer correction just like with a proper chronometer. The difference is your corrected time will be MORE accurate with your $50 digital than your $3k chro. And you don't even have to remember to wind it.


A $100 receiver and a piece of wire is all you need to receive time signals. Just sayin. a worthwhile addition to your toolkit, for comparing watch time to GMT. (UTC, actually.) If the big one comes and the radio goes silent, you still got your Casio or whatever, and now you know you can correct the time and trust it for navigation.


You can't always see the sky. When you can, sometimes you can't see the horizon. You can still shoot, if you know how, without a horizon with usable accuracy, but you must be able to see the body you are shooting. I have made Atlantic crossings where for the entire crossing not a single observation was possible. The knowledge of DR navigation and other parts of the navigator's toolkit would still have ensured a decent landfall without electronic navigation. So don't just learn celestial navigation. Learn NAVIGATION.

[fernandosmooth]

I will get to your post and digest it thoroughly but I have lot to think because this turned into a big thread and there is a lot of information.

[belizesailor]

Originally Posted by 14murs14

Actually I was chatting with someone about this and learned something new. Apparently they actually changed the orbit of several of the GPS satellites, which weren't as robust of a constellation as we have now, in order to ensure coverage over the middle east. It stands to reason that doing this would result in gaps elsewhere, probably right where you happened to be! Thanks for the story, made me learn something.

Could have been. This was a long time ago...the older units' software was not very robust...took them a long time to acquire under normal circumstances...changing the constellation would likely really confuse them for an extended time.

[belizesailor]

Originally Posted by SteveG58

Using a plumb bob on a board marked out with a protractor can give you a fairly accurate reading. Midday only I think!

A plumb line could certainly be used to determine LAN (Local Apparent Noon) as can a sextant.

From LAN you can derive latitude and of course approximate time.

[GrowleyMonster]

Originally Posted by belizesailor

A plumb line could certainly be used to determine LAN (Local Apparent Noon) as can a sextant.

From LAN you can derive latitude and of course approximate time.

Oops clicked wrong button.


Well, you can derive approximate time from LAN if you know your Longitude, yes. If you are not sure of your Longitude, then LAN observation will still leave you guessing what time it is. You need something to nail down either Longitude, from which you can easily derive time, or time, from which you can derive Longitude. One or the other. Unless you resort to more esoteric methods like Lunar Distance or other such astronomical/mathematical hocus pocus. Assumed Position or DR Position or Estimated Position is not going to cut it. Using such methods as a time check will put your DR and your non LAN LOPs off even more, and when you then use that erroneous Longitude to determine time by LAN the next day, you are off even more.

[longjonsilver]

OK, call me a luddite if you will, but i would like to find printable trig tables to the minute of accuracy. All i can find online is tables for beginners to the nearest degree or Branica tables. i have these tables in my handbook of chemistry, but would like something easier to read.
anyone?
jon

i'm currently studying this:
http://www.teacupnavigation.net/Cele...a_Teacup_v.pdf

[Jammer]

Originally Posted by longjonsilver

OK, call me a luddite if you will, but i would like to find printable trig tables to the minute of accuracy. All i can find online is tables for beginners to the nearest degree or Branica tables. i have these tables in my handbook of chemistry, but would like something easier to read.
anyone?
jon

i'm currently studying this:
http://www.teacupnavigation.net/Cele...a_Teacup_v.pdf

https://www.amazon.com/Interpolated-...garithm+tables


Most people with a modicum of software development expertise could construct a PDF file with the tables also, in the font and format of your choice.

[GrowleyMonster]

Originally Posted by longjonsilver

OK, call me a luddite if you will, but i would like to find printable trig tables to the minute of accuracy. All i can find online is tables for beginners to the nearest degree or Branica tables. i have these tables in my handbook of chemistry, but would like something easier to read.
anyone?
jon

i'm currently studying this:
http://www.teacupnavigation.net/Cele...a_Teacup_v.pdf

To the minute? A long table, that. But the power of the computer can benefit even the luddite mathematician. It would be fairly simple to write a program that would print out your tables faster than you could manually compute just one value. but why stop at the nearest whole minute of arc? You could even have the computer calculate and tabulate tables for angles to the nearest second. Or radians or fractional parts of radians to however many decimal places. Log tables, too. If you have access to CNC milling machinery you could even have your trig tables chiseled into stone tablets, and pave your monastery courtyard with them. From that point, you and your abacus should be able to handle things just fine.

[Jammer]

Originally Posted by longjonsilver

OK, call me a luddite if you will, but i would like to find printable trig tables to the minute of accuracy.f

Here you go. I was bored.



Table of sines and cosines in degrees and minutes.

sin +00' +01' +02' +03' +04' +05' +06' +07' +08' +09'
cos +10' +09' +08' +07' +06' +05' +04' +03' +02' +01'

=== Sine of 0 degrees / Cosine of 89 degrees ===
+00' +50 0.00000 0.00029 0.00058 0.00087 0.00116 0.00145 0.00175 0.00204 0.00233 0.00262
+10' +40 0.00291 0.00320 0.00349 0.00378 0.00407 0.00436 0.00465 0.00495 0.00524 0.00553
+20' +30 0.00582 0.00611 0.00640 0.00669 0.00698 0.00727 0.00756 0.00785 0.00814 0.00844
+30' +20 0.00873 0.00902 0.00931 0.00960 0.00989 0.01018 0.01047 0.01076 0.01105 0.01134
+40' +10 0.01164 0.01193 0.01222 0.01251 0.01280 0.01309 0.01338 0.01367 0.01396 0.01425
+50' +00 0.01454 0.01483 0.01513 0.01542 0.01571 0.01600 0.01629 0.01658 0.01687 0.01716
=== Sine of 1 degrees / Cosine of 88 degrees ===
+00' +50 0.01745 0.01774 0.01803 0.01832 0.01862 0.01891 0.01920 0.01949 0.01978 0.02007
+10' +40 0.02036 0.02065 0.02094 0.02123 0.02152 0.02181 0.02211 0.02240 0.02269 0.02298
+20' +30 0.02327 0.02356 0.02385 0.02414 0.02443 0.02472 0.02501 0.02530 0.02560 0.02589
+30' +20 0.02618 0.02647 0.02676 0.02705 0.02734 0.02763 0.02792 0.02821 0.02850 0.02879
+40' +10 0.02908 0.02938 0.02967 0.02996 0.03025 0.03054 0.03083 0.03112 0.03141 0.03170
+50' +00 0.03199 0.03228 0.03257 0.03286 0.03316 0.03345 0.03374 0.03403 0.03432 0.03461
=== Sine of 2 degrees / Cosine of 87 degrees ===
+00' +50 0.03490 0.03519 0.03548 0.03577 0.03606 0.03635 0.03664 0.03693 0.03723 0.03752
+10' +40 0.03781 0.03810 0.03839 0.03868 0.03897 0.03926 0.03955 0.03984 0.04013 0.04042
+20' +30 0.04071 0.04100 0.04129 0.04159 0.04188 0.04217 0.04246 0.04275 0.04304 0.04333
+30' +20 0.04362 0.04391 0.04420 0.04449 0.04478 0.04507 0.04536 0.04565 0.04594 0.04623
+40' +10 0.04653 0.04682 0.04711 0.04740 0.04769 0.04798 0.04827 0.04856 0.04885 0.04914
+50' +00 0.04943 0.04972 0.05001 0.05030 0.05059 0.05088 0.05117 0.05146 0.05175 0.05205
=== Sine of 3 degrees / Cosine of 86 degrees ===
+00' +50 0.05234 0.05263 0.05292 0.05321 0.05350 0.05379 0.05408 0.05437 0.05466 0.05495
+10' +40 0.05524 0.05553 0.05582 0.05611 0.05640 0.05669 0.05698 0.05727 0.05756 0.05785
+20' +30 0.05814 0.05844 0.05873 0.05902 0.05931 0.05960 0.05989 0.06018 0.06047 0.06076
+30' +20 0.06105 0.06134 0.06163 0.06192 0.06221 0.06250 0.06279 0.06308 0.06337 0.06366
+40' +10 0.06395 0.06424 0.06453 0.06482 0.06511 0.06540 0.06569 0.06598 0.06627 0.06656
+50' +00 0.06685 0.06714 0.06743 0.06773 0.06802 0.06831 0.06860 0.06889 0.06918 0.06947
=== Sine of 4 degrees / Cosine of 85 degrees ===
+00' +50 0.06976 0.07005 0.07034 0.07063 0.07092 0.07121 0.07150 0.07179 0.07208 0.07237
+10' +40 0.07266 0.07295 0.07324 0.07353 0.07382 0.07411 0.07440 0.07469 0.07498 0.07527
+20' +30 0.07556 0.07585 0.07614 0.07643 0.07672 0.07701 0.07730 0.07759 0.07788 0.07817
+30' +20 0.07846 0.07875 0.07904 0.07933 0.07962 0.07991 0.08020 0.08049 0.08078 0.08107
+40' +10 0.08136 0.08165 0.08194 0.08223 0.08252 0.08281 0.08310 0.08339 0.08368 0.08397
+50' +00 0.08426 0.08455 0.08484 0.08513 0.08542 0.08571 0.08600 0.08629 0.08658 0.08687
=== Sine of 5 degrees / Cosine of 84 degrees ===
+00' +50 0.08716 0.08745 0.08774 0.08803 0.08831 0.08860 0.08889 0.08918 0.08947 0.08976
+10' +40 0.09005 0.09034 0.09063 0.09092 0.09121 0.09150 0.09179 0.09208 0.09237 0.09266
+20' +30 0.09295 0.09324 0.09353 0.09382 0.09411 0.09440 0.09469 0.09498 0.09527 0.09556
+30' +20 0.09585 0.09614 0.09642 0.09671 0.09700 0.09729 0.09758 0.09787 0.09816 0.09845
+40' +10 0.09874 0.09903 0.09932 0.09961 0.09990 0.10019 0.10048 0.10077 0.10106 0.10135
+50' +00 0.10164 0.10192 0.10221 0.10250 0.10279 0.10308 0.10337 0.10366 0.10395 0.10424
=== Sine of 6 degrees / Cosine of 83 degrees ===
+00' +50 0.10453 0.10482 0.10511 0.10540 0.10569 0.10597 0.10626 0.10655 0.10684 0.10713
+10' +40 0.10742 0.10771 0.10800 0.10829 0.10858 0.10887 0.10916 0.10945 0.10973 0.11002
+20' +30 0.11031 0.11060 0.11089 0.11118 0.11147 0.11176 0.11205 0.11234 0.11263 0.11291
+30' +20 0.11320 0.11349 0.11378 0.11407 0.11436 0.11465 0.11494 0.11523 0.11552 0.11580
+40' +10 0.11609 0.11638 0.11667 0.11696 0.11725 0.11754 0.11783 0.11812 0.11840 0.11869
+50' +00 0.11898 0.11927 0.11956 0.11985 0.12014 0.12043 0.12071 0.12100 0.12129 0.12158
=== Sine of 7 degrees / Cosine of 82 degrees ===
+00' +50 0.12187 0.12216 0.12245 0.12274 0.12302 0.12331 0.12360 0.12389 0.12418 0.12447
+10' +40 0.12476 0.12504 0.12533 0.12562 0.12591 0.12620 0.12649 0.12678 0.12706 0.12735
+20' +30 0.12764 0.12793 0.12822 0.12851 0.12880 0.12908 0.12937 0.12966 0.12995 0.13024
+30' +20 0.13053 0.13081 0.13110 0.13139 0.13168 0.13197 0.13226 0.13254 0.13283 0.13312
+40' +10 0.13341 0.13370 0.13399 0.13427 0.13456 0.13485 0.13514 0.13543 0.13572 0.13600
+50' +00 0.13629 0.13658 0.13687 0.13716 0.13744 0.13773 0.13802 0.13831 0.13860 0.13889
=== Sine of 8 degrees / Cosine of 81 degrees ===
+00' +50 0.13917 0.13946 0.13975 0.14004 0.14033 0.14061 0.14090 0.14119 0.14148 0.14177
+10' +40 0.14205 0.14234 0.14263 0.14292 0.14320 0.14349 0.14378 0.14407 0.14436 0.14464
+20' +30 0.14493 0.14522 0.14551 0.14580 0.14608 0.14637 0.14666 0.14695 0.14723 0.14752
+30' +20 0.14781 0.14810 0.14838 0.14867 0.14896 0.14925 0.14954 0.14982 0.15011 0.15040
+40' +10 0.15069 0.15097 0.15126 0.15155 0.15184 0.15212 0.15241 0.15270 0.15299 0.15327
+50' +00 0.15356 0.15385 0.15414 0.15442 0.15471 0.15500 0.15529 0.15557 0.15586 0.15615
=== Sine of 9 degrees / Cosine of 80 degrees ===
+00' +50 0.15643 0.15672 0.15701 0.15730 0.15758 0.15787 0.15816 0.15845 0.15873 0.15902
+10' +40 0.15931 0.15959 0.15988 0.16017 0.16046 0.16074 0.16103 0.16132 0.16160 0.16189
+20' +30 0.16218 0.16246 0.16275 0.16304 0.16333 0.16361 0.16390 0.16419 0.16447 0.16476
+30' +20 0.16505 0.16533 0.16562 0.16591 0.16620 0.16648 0.16677 0.16706 0.16734 0.16763
+40' +10 0.16792 0.16820 0.16849 0.16878 0.16906 0.16935 0.16964 0.16992 0.17021 0.17050
+50' +00 0.17078 0.17107 0.17136 0.17164 0.17193 0.17222 0.17250 0.17279 0.17308 0.17336
=== Sine of 10 degrees / Cosine of 79 degrees ===
+00' +50 0.17365 0.17393 0.17422 0.17451 0.17479 0.17508 0.17537 0.17565 0.17594 0.17623
+10' +40 0.17651 0.17680 0.17708 0.17737 0.17766 0.17794 0.17823 0.17852 0.17880 0.17909
+20' +30 0.17937 0.17966 0.17995 0.18023 0.18052 0.18081 0.18109 0.18138 0.18166 0.18195
+30' +20 0.18224 0.18252 0.18281 0.18309 0.18338 0.18367 0.18395 0.18424 0.18452 0.18481
+40' +10 0.18509 0.18538 0.18567 0.18595 0.18624 0.18652 0.18681 0.18710 0.18738 0.18767
+50' +00 0.18795 0.18824 0.18852 0.18881 0.18910 0.18938 0.18967 0.18995 0.19024 0.19052
=== Sine of 11 degrees / Cosine of 78 degrees ===
+00' +50 0.19081 0.19109 0.19138 0.19167 0.19195 0.19224 0.19252 0.19281 0.19309 0.19338
+10' +40 0.19366 0.19395 0.19423 0.19452 0.19481 0.19509 0.19538 0.19566 0.19595 0.19623
+20' +30 0.19652 0.19680 0.19709 0.19737 0.19766 0.19794 0.19823 0.19851 0.19880 0.19908
+30' +20 0.19937 0.19965 0.19994 0.20022 0.20051 0.20079 0.20108 0.20136 0.20165 0.20193
+40' +10 0.20222 0.20250 0.20279 0.20307 0.20336 0.20364 0.20393 0.20421 0.20450 0.20478
+50' +00 0.20507 0.20535 0.20563 0.20592 0.20620 0.20649 0.20677 0.20706 0.20734 0.20763
=== Sine of 12 degrees / Cosine of 77 degrees ===
+00' +50 0.20791 0.20820 0.20848 0.20877 0.20905 0.20933 0.20962 0.20990 0.21019 0.21047
+10' +40 0.21076 0.21104 0.21132 0.21161 0.21189 0.21218 0.21246 0.21275 0.21303 0.21331
+20' +30 0.21360 0.21388 0.21417 0.21445 0.21474 0.21502 0.21530 0.21559 0.21587 0.21616
+30' +20 0.21644 0.21672 0.21701 0.21729 0.21758 0.21786 0.21814 0.21843 0.21871 0.21899
+40' +10 0.21928 0.21956 0.21985 0.22013 0.22041 0.22070 0.22098 0.22126 0.22155 0.22183
+50' +00 0.22212 0.22240 0.22268 0.22297 0.22325 0.22353 0.22382 0.22410 0.22438 0.22467
=== Sine of 13 degrees / Cosine of 76 degrees ===
+00' +50 0.22495 0.22523 0.22552 0.22580 0.22608 0.22637 0.22665 0.22693 0.22722 0.22750
+10' +40 0.22778 0.22807 0.22835 0.22863 0.22892 0.22920 0.22948 0.22977 0.23005 0.23033
+20' +30 0.23062 0.23090 0.23118 0.23146 0.23175 0.23203 0.23231 0.23260 0.23288 0.23316
+30' +20 0.23345 0.23373 0.23401 0.23429 0.23458 0.23486 0.23514 0.23542 0.23571 0.23599
+40' +10 0.23627 0.23656 0.23684 0.23712 0.23740 0.23769 0.23797 0.23825 0.23853 0.23882
+50' +00 0.23910 0.23938 0.23966 0.23995 0.24023 0.24051 0.24079 0.24108 0.24136 0.24164
=== Sine of 14 degrees / Cosine of 75 degrees ===
+00' +50 0.24192 0.24220 0.24249 0.24277 0.24305 0.24333 0.24362 0.24390 0.24418 0.24446
+10' +40 0.24474 0.24503 0.24531 0.24559 0.24587 0.24615 0.24644 0.24672 0.24700 0.24728
+20' +30 0.24756 0.24784 0.24813 0.24841 0.24869 0.24897 0.24925 0.24954 0.24982 0.25010
+30' +20 0.25038 0.25066 0.25094 0.25122 0.25151 0.25179 0.25207 0.25235 0.25263 0.25291
+40' +10 0.25320 0.25348 0.25376 0.25404 0.25432 0.25460 0.25488 0.25516 0.25545 0.25573
+50' +00 0.25601 0.25629 0.25657 0.25685 0.25713 0.25741 0.25769 0.25798 0.25826 0.25854
=== Sine of 15 degrees / Cosine of 74 degrees ===
+00' +50 0.25882 0.25910 0.25938 0.25966 0.25994 0.26022 0.26050 0.26079 0.26107 0.26135
+10' +40 0.26163 0.26191 0.26219 0.26247 0.26275 0.26303 0.26331 0.26359 0.26387 0.26415
+20' +30 0.26443 0.26471 0.26500 0.26528 0.26556 0.26584 0.26612 0.26640 0.26668 0.26696
+30' +20 0.26724 0.26752 0.26780 0.26808 0.26836 0.26864 0.26892 0.26920 0.26948 0.26976
+40' +10 0.27004 0.27032 0.27060 0.27088 0.27116 0.27144 0.27172 0.27200 0.27228 0.27256
+50' +00 0.27284 0.27312 0.27340 0.27368 0.27396 0.27424 0.27452 0.27480 0.27508 0.27536
=== Sine of 16 degrees / Cosine of 73 degrees ===
+00' +50 0.27564 0.27592 0.27620 0.27648 0.27676 0.27704 0.27731 0.27759 0.27787 0.27815
+10' +40 0.27843 0.27871 0.27899 0.27927 0.27955 0.27983 0.28011 0.28039 0.28067 0.28095
+20' +30 0.28123 0.28150 0.28178 0.28206 0.28234 0.28262 0.28290 0.28318 0.28346 0.28374
+30' +20 0.28402 0.28429 0.28457 0.28485 0.28513 0.28541 0.28569 0.28597 0.28625 0.28652
+40' +10 0.28680 0.28708 0.28736 0.28764 0.28792 0.28820 0.28847 0.28875 0.28903 0.28931
+50' +00 0.28959 0.28987 0.29015 0.29042 0.29070 0.29098 0.29126 0.29154 0.29182 0.29209
=== Sine of 17 degrees / Cosine of 72 degrees ===
+00' +50 0.29237 0.29265 0.29293 0.29321 0.29348 0.29376 0.29404 0.29432 0.29460 0.29487
+10' +40 0.29515 0.29543 0.29571 0.29599 0.29626 0.29654 0.29682 0.29710 0.29737 0.29765
+20' +30 0.29793 0.29821 0.29849 0.29876 0.29904 0.29932 0.29960 0.29987 0.30015 0.30043
+30' +20 0.30071 0.30098 0.30126 0.30154 0.30182 0.30209 0.30237 0.30265 0.30292 0.30320
+40' +10 0.30348 0.30376 0.30403 0.30431 0.30459 0.30486 0.30514 0.30542 0.30570 0.30597
+50' +00 0.30625 0.30653 0.30680 0.30708 0.30736 0.30763 0.30791 0.30819 0.30846 0.30874
=== Sine of 18 degrees / Cosine of 71 degrees ===
+00' +50 0.30902 0.30929 0.30957 0.30985 0.31012 0.31040 0.31068 0.31095 0.31123 0.31151
+10' +40 0.31178 0.31206 0.31233 0.31261 0.31289 0.31316 0.31344 0.31372 0.31399 0.31427
+20' +30 0.31454 0.31482 0.31510 0.31537 0.31565 0.31593 0.31620 0.31648 0.31675 0.31703
+30' +20 0.31730 0.31758 0.31786 0.31813 0.31841 0.31868 0.31896 0.31923 0.31951 0.31979
+40' +10 0.32006 0.32034 0.32061 0.32089 0.32116 0.32144 0.32171 0.32199 0.32227 0.32254
+50' +00 0.32282 0.32309 0.32337 0.32364 0.32392 0.32419 0.32447 0.32474 0.32502 0.32529
=== Sine of 19 degrees / Cosine of 70 degrees ===
+00' +50 0.32557 0.32584 0.32612 0.32639 0.32667 0.32694 0.32722 0.32749 0.32777 0.32804
+10' +40 0.32832 0.32859 0.32887 0.32914 0.32942 0.32969 0.32997 0.33024 0.33051 0.33079
+20' +30 0.33106 0.33134 0.33161 0.33189 0.33216 0.33244 0.33271 0.33298 0.33326 0.33353
+30' +20 0.33381 0.33408 0.33436 0.33463 0.33490 0.33518 0.33545 0.33573 0.33600 0.33627
+40' +10 0.33655 0.33682 0.33710 0.33737 0.33764 0.33792 0.33819 0.33846 0.33874 0.33901
+50' +00 0.33929 0.33956 0.33983 0.34011 0.34038 0.34065 0.34093 0.34120 0.34147 0.34175
=== Sine of 20 degrees / Cosine of 69 degrees ===
+00' +50 0.34202 0.34229 0.34257 0.34284 0.34311 0.34339 0.34366 0.34393 0.34421 0.34448
+10' +40 0.34475 0.34503 0.34530 0.34557 0.34584 0.34612 0.34639 0.34666 0.34694 0.34721
+20' +30 0.34748 0.34775 0.34803 0.34830 0.34857 0.34884 0.34912 0.34939 0.34966 0.34993
+30' +20 0.35021 0.35048 0.35075 0.35102 0.35130 0.35157 0.35184 0.35211 0.35239 0.35266
+40' +10 0.35293 0.35320 0.35347 0.35375 0.35402 0.35429 0.35456 0.35484 0.35511 0.35538
+50' +00 0.35565 0.35592 0.35619 0.35647 0.35674 0.35701 0.35728 0.35755 0.35782 0.35810
=== Sine of 21 degrees / Cosine of 68 degrees ===
+00' +50 0.35837 0.35864 0.35891 0.35918 0.35945 0.35973 0.36000 0.36027 0.36054 0.36081
+10' +40 0.36108 0.36135 0.36162 0.36190 0.36217 0.36244 0.36271 0.36298 0.36325 0.36352
+20' +30 0.36379 0.36406 0.36434 0.36461 0.36488 0.36515 0.36542 0.36569 0.36596 0.36623
+30' +20 0.36650 0.36677 0.36704 0.36731 0.36758 0.36785 0.36812 0.36839 0.36867 0.36894
+40' +10 0.36921 0.36948 0.36975 0.37002 0.37029 0.37056 0.37083 0.37110 0.37137 0.37164
+50' +00 0.37191 0.37218 0.37245 0.37272 0.37299 0.37326 0.37353 0.37380 0.37407 0.37434
=== Sine of 22 degrees / Cosine of 67 degrees ===
+00' +50 0.37461 0.37488 0.37515 0.37542 0.37569 0.37595 0.37622 0.37649 0.37676 0.37703
+10' +40 0.37730 0.37757 0.37784 0.37811 0.37838 0.37865 0.37892 0.37919 0.37946 0.37973
+20' +30 0.37999 0.38026 0.38053 0.38080 0.38107 0.38134 0.38161 0.38188 0.38215 0.38241
+30' +20 0.38268 0.38295 0.38322 0.38349 0.38376 0.38403 0.38430 0.38456 0.38483 0.38510
+40' +10 0.38537 0.38564 0.38591 0.38617 0.38644 0.38671 0.38698 0.38725 0.38752 0.38778
+50' +00 0.38805 0.38832 0.38859 0.38886 0.38912 0.38939 0.38966 0.38993 0.39020 0.39046
=== Sine of 23 degrees / Cosine of 66 degrees ===
+00' +50 0.39073 0.39100 0.39127 0.39153 0.39180 0.39207 0.39234 0.39260 0.39287 0.39314
+10' +40 0.39341 0.39367 0.39394 0.39421 0.39448 0.39474 0.39501 0.39528 0.39555 0.39581
+20' +30 0.39608 0.39635 0.39661 0.39688 0.39715 0.39741 0.39768 0.39795 0.39822 0.39848
+30' +20 0.39875 0.39902 0.39928 0.39955 0.39982 0.40008 0.40035 0.40062 0.40088 0.40115
+40' +10 0.40141 0.40168 0.40195 0.40221 0.40248 0.40275 0.40301 0.40328 0.40355 0.40381
+50' +00 0.40408 0.40434 0.40461 0.40488 0.40514 0.40541 0.40567 0.40594 0.40621 0.40647
=== Sine of 24 degrees / Cosine of 65 degrees ===
+00' +50 0.40674 0.40700 0.40727 0.40753 0.40780 0.40806 0.40833 0.40860 0.40886 0.40913
+10' +40 0.40939 0.40966 0.40992 0.41019 0.41045 0.41072 0.41098 0.41125 0.41151 0.41178
+20' +30 0.41204 0.41231 0.41257 0.41284 0.41310 0.41337 0.41363 0.41390 0.41416 0.41443
+30' +20 0.41469 0.41496 0.41522 0.41549 0.41575 0.41602 0.41628 0.41655 0.41681 0.41707
+40' +10 0.41734 0.41760 0.41787 0.41813 0.41840 0.41866 0.41892 0.41919 0.41945 0.41972
+50' +00 0.41998 0.42024 0.42051 0.42077 0.42104 0.42130 0.42156 0.42183 0.42209 0.42235
=== Sine of 25 degrees / Cosine of 64 degrees ===
+00' +50 0.42262 0.42288 0.42315 0.42341 0.42367 0.42394 0.42420 0.42446 0.42473 0.42499
+10' +40 0.42525 0.42552 0.42578 0.42604 0.42631 0.42657 0.42683 0.42709 0.42736 0.42762
+20' +30 0.42788 0.42815 0.42841 0.42867 0.42894 0.42920 0.42946 0.42972 0.42999 0.43025
+30' +20 0.43051 0.43077 0.43104 0.43130 0.43156 0.43182 0.43209 0.43235 0.43261 0.43287
+40' +10 0.43313 0.43340 0.43366 0.43392 0.43418 0.43445 0.43471 0.43497 0.43523 0.43549
+50' +00 0.43575 0.43602 0.43628 0.43654 0.43680 0.43706 0.43733 0.43759 0.43785 0.43811
=== Sine of 26 degrees / Cosine of 63 degrees ===
+00' +50 0.43837 0.43863 0.43889 0.43916 0.43942 0.43968 0.43994 0.44020 0.44046 0.44072
+10' +40 0.44098 0.44124 0.44151 0.44177 0.44203 0.44229 0.44255 0.44281 0.44307 0.44333
+20' +30 0.44359 0.44385 0.44411 0.44437 0.44464 0.44490 0.44516 0.44542 0.44568 0.44594
+30' +20 0.44620 0.44646 0.44672 0.44698 0.44724 0.44750 0.44776 0.44802 0.44828 0.44854
+40' +10 0.44880 0.44906 0.44932 0.44958 0.44984 0.45010 0.45036 0.45062 0.45088 0.45114
+50' +00 0.45140 0.45166 0.45192 0.45218 0.45243 0.45269 0.45295 0.45321 0.45347 0.45373
=== Sine of 27 degrees / Cosine of 62 degrees ===
+00' +50 0.45399 0.45425 0.45451 0.45477 0.45503 0.45529 0.45554 0.45580 0.45606 0.45632
+10' +40 0.45658 0.45684 0.45710 0.45736 0.45762 0.45787 0.45813 0.45839 0.45865 0.45891
+20' +30 0.45917 0.45942 0.45968 0.45994 0.46020 0.46046 0.46072 0.46097 0.46123 0.46149
+30' +20 0.46175 0.46201 0.46226 0.46252 0.46278 0.46304 0.46330 0.46355 0.46381 0.46407
+40' +10 0.46433 0.46458 0.46484 0.46510 0.46536 0.46561 0.46587 0.46613 0.46639 0.46664
+50' +00 0.46690 0.46716 0.46742 0.46767 0.46793 0.46819 0.46844 0.46870 0.46896 0.46921
=== Sine of 28 degrees / Cosine of 61 degrees ===
+00' +50 0.46947 0.46973 0.46999 0.47024 0.47050 0.47076 0.47101 0.47127 0.47153 0.47178
+10' +40 0.47204 0.47229 0.47255 0.47281 0.47306 0.47332 0.47358 0.47383 0.47409 0.47434
+20' +30 0.47460 0.47486 0.47511 0.47537 0.47562 0.47588 0.47614 0.47639 0.47665 0.47690
+30' +20 0.47716 0.47741 0.47767 0.47793 0.47818 0.47844 0.47869 0.47895 0.47920 0.47946
+40' +10 0.47971 0.47997 0.48022 0.48048 0.48073 0.48099 0.48124 0.48150 0.48175 0.48201
+50' +00 0.48226 0.48252 0.48277 0.48303 0.48328 0.48354 0.48379 0.48405 0.48430 0.48456
=== Sine of 29 degrees / Cosine of 60 degrees ===
+00' +50 0.48481 0.48506 0.48532 0.48557 0.48583 0.48608 0.48634 0.48659 0.48684 0.48710
+10' +40 0.48735 0.48761 0.48786 0.48811 0.48837 0.48862 0.48888 0.48913 0.48938 0.48964
+20' +30 0.48989 0.49014 0.49040 0.49065 0.49090 0.49116 0.49141 0.49166 0.49192 0.49217
+30' +20 0.49242 0.49268 0.49293 0.49318 0.49344 0.49369 0.49394 0.49419 0.49445 0.49470
+40' +10 0.49495 0.49521 0.49546 0.49571 0.49596 0.49622 0.49647 0.49672 0.49697 0.49723
+50' +00 0.49748 0.49773 0.49798 0.49824 0.49849 0.49874 0.49899 0.49924 0.49950 0.49975
=== Sine of 30 degrees / Cosine of 59 degrees ===
+00' +50 0.50000 0.50025 0.50050 0.50076 0.50101 0.50126 0.50151 0.50176 0.50201 0.50227
+10' +40 0.50252 0.50277 0.50302 0.50327 0.50352 0.50377 0.50403 0.50428 0.50453 0.50478
+20' +30 0.50503 0.50528 0.50553 0.50578 0.50603 0.50628 0.50654 0.50679 0.50704 0.50729
+30' +20 0.50754 0.50779 0.50804 0.50829 0.50854 0.50879 0.50904 0.50929 0.50954 0.50979
+40' +10 0.51004 0.51029 0.51054 0.51079 0.51104 0.51129 0.51154 0.51179 0.51204 0.51229
+50' +00 0.51254 0.51279 0.51304 0.51329 0.51354 0.51379 0.51404 0.51429 0.51454 0.51479
=== Sine of 31 degrees / Cosine of 58 degrees ===
+00' +50 0.51504 0.51529 0.51554 0.51579 0.51604 0.51628 0.51653 0.51678 0.51703 0.51728
+10' +40 0.51753 0.51778 0.51803 0.51828 0.51852 0.51877 0.51902 0.51927 0.51952 0.51977
+20' +30 0.52002 0.52026 0.52051 0.52076 0.52101 0.52126 0.52151 0.52175 0.52200 0.52225
+30' +20 0.52250 0.52275 0.52299 0.52324 0.52349 0.52374 0.52399 0.52423 0.52448 0.52473
+40' +10 0.52498 0.52522 0.52547 0.52572 0.52597 0.52621 0.52646 0.52671 0.52696 0.52720
+50' +00 0.52745 0.52770 0.52794 0.52819 0.52844 0.52869 0.52893 0.52918 0.52943 0.52967
=== Sine of 32 degrees / Cosine of 57 degrees ===
+00' +50 0.52992 0.53017 0.53041 0.53066 0.53091 0.53115 0.53140 0.53164 0.53189 0.53214
+10' +40 0.53238 0.53263 0.53288 0.53312 0.53337 0.53361 0.53386 0.53411 0.53435 0.53460
+20' +30 0.53484 0.53509 0.53534 0.53558 0.53583 0.53607 0.53632 0.53656 0.53681 0.53705
+30' +20 0.53730 0.53754 0.53779 0.53804 0.53828 0.53853 0.53877 0.53902 0.53926 0.53951
+40' +10 0.53975 0.54000 0.54024 0.54049 0.54073 0.54097 0.54122 0.54146 0.54171 0.54195
+50' +00 0.54220 0.54244 0.54269 0.54293 0.54317 0.54342 0.54366 0.54391 0.54415 0.54440
=== Sine of 33 degrees / Cosine of 56 degrees ===
+00' +50 0.54464 0.54488 0.54513 0.54537 0.54561 0.54586 0.54610 0.54635 0.54659 0.54683
+10' +40 0.54708 0.54732 0.54756 0.54781 0.54805 0.54829 0.54854 0.54878 0.54902 0.54927
+20' +30 0.54951 0.54975 0.54999 0.55024 0.55048 0.55072 0.55097 0.55121 0.55145 0.55169
+30' +20 0.55194 0.55218 0.55242 0.55266 0.55291 0.55315 0.55339 0.55363 0.55388 0.55412
+40' +10 0.55436 0.55460 0.55484 0.55509 0.55533 0.55557 0.55581 0.55605 0.55630 0.55654
+50' +00 0.55678 0.55702 0.55726 0.55750 0.55775 0.55799 0.55823 0.55847 0.55871 0.55895
=== Sine of 34 degrees / Cosine of 55 degrees ===
+00' +50 0.55919 0.55943 0.55968 0.55992 0.56016 0.56040 0.56064 0.56088 0.56112 0.56136
+10' +40 0.56160 0.56184 0.56208 0.56232 0.56256 0.56280 0.56305 0.56329 0.56353 0.56377
+20' +30 0.56401 0.56425 0.56449 0.56473 0.56497 0.56521 0.56545 0.56569 0.56593 0.56617
+30' +20 0.56641 0.56665 0.56689 0.56713 0.56736 0.56760 0.56784 0.56808 0.56832 0.56856
+40' +10 0.56880 0.56904 0.56928 0.56952 0.56976 0.57000 0.57024 0.57047 0.57071 0.57095
+50' +00 0.57119 0.57143 0.57167 0.57191 0.57215 0.57238 0.57262 0.57286 0.57310 0.57334
=== Sine of 35 degrees / Cosine of 54 degrees ===
+00' +50 0.57358 0.57381 0.57405 0.57429 0.57453 0.57477 0.57501 0.57524 0.57548 0.57572
+10' +40 0.57596 0.57619 0.57643 0.57667 0.57691 0.57715 0.57738 0.57762 0.57786 0.57810
+20' +30 0.57833 0.57857 0.57881 0.57904 0.57928 0.57952 0.57976 0.57999 0.58023 0.58047
+30' +20 0.58070 0.58094 0.58118 0.58141 0.58165 0.58189 0.58212 0.58236 0.58260 0.58283
+40' +10 0.58307 0.58330 0.58354 0.58378 0.58401 0.58425 0.58449 0.58472 0.58496 0.58519
+50' +00 0.58543 0.58567 0.58590 0.58614 0.58637 0.58661 0.58684 0.58708 0.58731 0.58755
=== Sine of 36 degrees / Cosine of 53 degrees ===
+00' +50 0.58779 0.58802 0.58826 0.58849 0.58873 0.58896 0.58920 0.58943 0.58967 0.58990
+10' +40 0.59014 0.59037 0.59061 0.59084 0.59108 0.59131 0.59154 0.59178 0.59201 0.59225
+20' +30 0.59248 0.59272 0.59295 0.59318 0.59342 0.59365 0.59389 0.59412 0.59436 0.59459
+30' +20 0.59482 0.59506 0.59529 0.59552 0.59576 0.59599 0.59622 0.59646 0.59669 0.59693
+40' +10 0.59716 0.59739 0.59763 0.59786 0.59809 0.59832 0.59856 0.59879 0.59902 0.59926
+50' +00 0.59949 0.59972 0.59995 0.60019 0.60042 0.60065 0.60089 0.60112 0.60135 0.60158
=== Sine of 37 degrees / Cosine of 52 degrees ===
+00' +50 0.60182 0.60205 0.60228 0.60251 0.60274 0.60298 0.60321 0.60344 0.60367 0.60390
+10' +40 0.60414 0.60437 0.60460 0.60483 0.60506 0.60529 0.60553 0.60576 0.60599 0.60622
+20' +30 0.60645 0.60668 0.60691 0.60714 0.60738 0.60761 0.60784 0.60807 0.60830 0.60853
+30' +20 0.60876 0.60899 0.60922 0.60945 0.60968 0.60991 0.61015 0.61038 0.61061 0.61084
+40' +10 0.61107 0.61130 0.61153 0.61176 0.61199 0.61222 0.61245 0.61268 0.61291 0.61314
+50' +00 0.61337 0.61360 0.61383 0.61406 0.61429 0.61451 0.61474 0.61497 0.61520 0.61543
=== Sine of 38 degrees / Cosine of 51 degrees ===
+00' +50 0.61566 0.61589 0.61612 0.61635 0.61658 0.61681 0.61704 0.61726 0.61749 0.61772
+10' +40 0.61795 0.61818 0.61841 0.61864 0.61887 0.61909 0.61932 0.61955 0.61978 0.62001
+20' +30 0.62024 0.62046 0.62069 0.62092 0.62115 0.62138 0.62160 0.62183 0.62206 0.62229
+30' +20 0.62251 0.62274 0.62297 0.62320 0.62342 0.62365 0.62388 0.62411 0.62433 0.62456
+40' +10 0.62479 0.62502 0.62524 0.62547 0.62570 0.62592 0.62615 0.62638 0.62660 0.62683
+50' +00 0.62706 0.62728 0.62751 0.62774 0.62796 0.62819 0.62842 0.62864 0.62887 0.62909
=== Sine of 39 degrees / Cosine of 50 degrees ===
+00' +50 0.62932 0.62955 0.62977 0.63000 0.63022 0.63045 0.63068 0.63090 0.63113 0.63135
+10' +40 0.63158 0.63180 0.63203 0.63225 0.63248 0.63271 0.63293 0.63316 0.63338 0.63361
+20' +30 0.63383 0.63406 0.63428 0.63451 0.63473 0.63496 0.63518 0.63540 0.63563 0.63585
+30' +20 0.63608 0.63630 0.63653 0.63675 0.63698 0.63720 0.63742 0.63765 0.63787 0.63810
+40' +10 0.63832 0.63854 0.63877 0.63899 0.63922 0.63944 0.63966 0.63989 0.64011 0.64033
+50' +00 0.64056 0.64078 0.64100 0.64123 0.64145 0.64167 0.64190 0.64212 0.64234 0.64256
=== Sine of 40 degrees / Cosine of 49 degrees ===
+00' +50 0.64279 0.64301 0.64323 0.64346 0.64368 0.64390 0.64412 0.64435 0.64457 0.64479
+10' +40 0.64501 0.64524 0.64546 0.64568 0.64590 0.64612 0.64635 0.64657 0.64679 0.64701
+20' +30 0.64723 0.64746 0.64768 0.64790 0.64812 0.64834 0.64856 0.64878 0.64901 0.64923
+30' +20 0.64945 0.64967 0.64989 0.65011 0.65033 0.65055 0.65077 0.65100 0.65122 0.65144
+40' +10 0.65166 0.65188 0.65210 0.65232 0.65254 0.65276 0.65298 0.65320 0.65342 0.65364
+50' +00 0.65386 0.65408 0.65430 0.65452 0.65474 0.65496 0.65518 0.65540 0.65562 0.65584
=== Sine of 41 degrees / Cosine of 48 degrees ===
+00' +50 0.65606 0.65628 0.65650 0.65672 0.65694 0.65716 0.65738 0.65759 0.65781 0.65803
+10' +40 0.65825 0.65847 0.65869 0.65891 0.65913 0.65935 0.65956 0.65978 0.66000 0.66022
+20' +30 0.66044 0.66066 0.66088 0.66109 0.66131 0.66153 0.66175 0.66197 0.66218 0.66240
+30' +20 0.66262 0.66284 0.66306 0.66327 0.66349 0.66371 0.66393 0.66414 0.66436 0.66458
+40' +10 0.66480 0.66501 0.66523 0.66545 0.66566 0.66588 0.66610 0.66632 0.66653 0.66675
+50' +00 0.66697 0.66718 0.66740 0.66762 0.66783 0.66805 0.66827 0.66848 0.66870 0.66891
=== Sine of 42 degrees / Cosine of 47 degrees ===
+00' +50 0.66913 0.66935 0.66956 0.66978 0.66999 0.67021 0.67043 0.67064 0.67086 0.67107
+10' +40 0.67129 0.67151 0.67172 0.67194 0.67215 0.67237 0.67258 0.67280 0.67301 0.67323
+20' +30 0.67344 0.67366 0.67387 0.67409 0.67430 0.67452 0.67473 0.67495 0.67516 0.67538
+30' +20 0.67559 0.67580 0.67602 0.67623 0.67645 0.67666 0.67688 0.67709 0.67730 0.67752
+40' +10 0.67773 0.67795 0.67816 0.67837 0.67859 0.67880 0.67901 0.67923 0.67944 0.67965
+50' +00 0.67987 0.68008 0.68029 0.68051 0.68072 0.68093 0.68115 0.68136 0.68157 0.68179
=== Sine of 43 degrees / Cosine of 46 degrees ===
+00' +50 0.68200 0.68221 0.68242 0.68264 0.68285 0.68306 0.68327 0.68349 0.68370 0.68391
+10' +40 0.68412 0.68434 0.68455 0.68476 0.68497 0.68518 0.68539 0.68561 0.68582 0.68603
+20' +30 0.68624 0.68645 0.68666 0.68688 0.68709 0.68730 0.68751 0.68772 0.68793 0.68814
+30' +20 0.68835 0.68857 0.68878 0.68899 0.68920 0.68941 0.68962 0.68983 0.69004 0.69025
+40' +10 0.69046 0.69067 0.69088 0.69109 0.69130 0.69151 0.69172 0.69193 0.69214 0.69235
+50' +00 0.69256 0.69277 0.69298 0.69319 0.69340 0.69361 0.69382 0.69403 0.69424 0.69445
=== Sine of 44 degrees / Cosine of 45 degrees ===
+00' +50 0.69466 0.69487 0.69508 0.69529 0.69549 0.69570 0.69591 0.69612 0.69633 0.69654
+10' +40 0.69675 0.69696 0.69717 0.69737 0.69758 0.69779 0.69800 0.69821 0.69842 0.69862
+20' +30 0.69883 0.69904 0.69925 0.69946 0.69966 0.69987 0.70008 0.70029 0.70049 0.70070
+30' +20 0.70091 0.70112 0.70132 0.70153 0.70174 0.70195 0.70215 0.70236 0.70257 0.70277
+40' +10 0.70298 0.70319 0.70339 0.70360 0.70381 0.70401 0.70422 0.70443 0.70463 0.70484
+50' +00 0.70505 0.70525 0.70546 0.70567 0.70587 0.70608 0.70628 0.70649 0.70670 0.70690
=== Sine of 45 degrees / Cosine of 44 degrees ===
+00' +50 0.70711 0.70731 0.70752 0.70772 0.70793 0.70813 0.70834 0.70855 0.70875 0.70896
+10' +40 0.70916 0.70937 0.70957 0.70978 0.70998 0.71019 0.71039 0.71059 0.71080 0.71100
+20' +30 0.71121 0.71141 0.71162 0.71182 0.71203 0.71223 0.71243 0.71264 0.71284 0.71305
+30' +20 0.71325 0.71345 0.71366 0.71386 0.71407 0.71427 0.71447 0.71468 0.71488 0.71508
+40' +10 0.71529 0.71549 0.71569 0.71590 0.71610 0.71630 0.71650 0.71671 0.71691 0.71711
+50' +00 0.71732 0.71752 0.71772 0.71792 0.71813 0.71833 0.71853 0.71873 0.71894 0.71914
=== Sine of 46 degrees / Cosine of 43 degrees ===
+00' +50 0.71934 0.71954 0.71974 0.71995 0.72015 0.72035 0.72055 0.72075 0.72095 0.72116
+10' +40 0.72136 0.72156 0.72176 0.72196 0.72216 0.72236 0.72257 0.72277 0.72297 0.72317
+20' +30 0.72337 0.72357 0.72377 0.72397 0.72417 0.72437 0.72457 0.72477 0.72497 0.72517
+30' +20 0.72537 0.72557 0.72577 0.72597 0.72617 0.72637 0.72657 0.72677 0.72697 0.72717
+40' +10 0.72737 0.72757 0.72777 0.72797 0.72817 0.72837 0.72857 0.72877 0.72897 0.72917
+50' +00 0.72937 0.72957 0.72976 0.72996 0.73016 0.73036 0.73056 0.73076 0.73096 0.73116
=== Sine of 47 degrees / Cosine of 42 degrees ===
+00' +50 0.73135 0.73155 0.73175 0.73195 0.73215 0.73234 0.73254 0.73274 0.73294 0.73314
+10' +40 0.73333 0.73353 0.73373 0.73393 0.73413 0.73432 0.73452 0.73472 0.73491 0.73511
+20' +30 0.73531 0.73551 0.73570 0.73590 0.73610 0.73629 0.73649 0.73669 0.73688 0.73708
+30' +20 0.73728 0.73747 0.73767 0.73787 0.73806 0.73826 0.73846 0.73865 0.73885 0.73904
+40' +10 0.73924 0.73944 0.73963 0.73983 0.74002 0.74022 0.74041 0.74061 0.74080 0.74100
+50' +00 0.74120 0.74139 0.74159 0.74178 0.74198 0.74217 0.74237 0.74256 0.74276 0.74295
=== Sine of 48 degrees / Cosine of 41 degrees ===
+00' +50 0.74314 0.74334 0.74353 0.74373 0.74392 0.74412 0.74431 0.74451 0.74470 0.74489
+10' +40 0.74509 0.74528 0.74548 0.74567 0.74586 0.74606 0.74625 0.74644 0.74664 0.74683
+20' +30 0.74703 0.74722 0.74741 0.74760 0.74780 0.74799 0.74818 0.74838 0.74857 0.74876
+30' +20 0.74896 0.74915 0.74934 0.74953 0.74973 0.74992 0.75011 0.75030 0.75050 0.75069
+40' +10 0.75088 0.75107 0.75126 0.75146 0.75165 0.75184 0.75203 0.75222 0.75241 0.75261
+50' +00 0.75280 0.75299 0.75318 0.75337 0.75356 0.75375 0.75395 0.75414 0.75433 0.75452
=== Sine of 49 degrees / Cosine of 40 degrees ===
+00' +50 0.75471 0.75490 0.75509 0.75528 0.75547 0.75566 0.75585 0.75604 0.75623 0.75642
+10' +40 0.75661 0.75680 0.75700 0.75719 0.75738 0.75756 0.75775 0.75794 0.75813 0.75832
+20' +30 0.75851 0.75870 0.75889 0.75908 0.75927 0.75946 0.75965 0.75984 0.76003 0.76022
+30' +20 0.76041 0.76059 0.76078 0.76097 0.76116 0.76135 0.76154 0.76173 0.76192 0.76210
+40' +10 0.76229 0.76248 0.76267 0.76286 0.76304 0.76323 0.76342 0.76361 0.76380 0.76398
+50' +00 0.76417 0.76436 0.76455 0.76473 0.76492 0.76511 0.76530 0.76548 0.76567 0.76586
=== Sine of 50 degrees / Cosine of 39 degrees ===
+00' +50 0.76604 0.76623 0.76642 0.76661 0.76679 0.76698 0.76717 0.76735 0.76754 0.76772
+10' +40 0.76791 0.76810 0.76828 0.76847 0.76866 0.76884 0.76903 0.76921 0.76940 0.76959
+20' +30 0.76977 0.76996 0.77014 0.77033 0.77051 0.77070 0.77088 0.77107 0.77125 0.77144
+30' +20 0.77162 0.77181 0.77199 0.77218 0.77236 0.77255 0.77273 0.77292 0.77310 0.77329
+40' +10 0.77347 0.77366 0.77384 0.77402 0.77421 0.77439 0.77458 0.77476 0.77494 0.77513
+50' +00 0.77531 0.77550 0.77568 0.77586 0.77605 0.77623 0.77641 0.77660 0.77678 0.77696
=== Sine of 51 degrees / Cosine of 38 degrees ===
+00' +50 0.77715 0.77733 0.77751 0.77769 0.77788 0.77806 0.77824 0.77843 0.77861 0.77879
+10' +40 0.77897 0.77916 0.77934 0.77952 0.77970 0.77988 0.78007 0.78025 0.78043 0.78061
+20' +30 0.78079 0.78098 0.78116 0.78134 0.78152 0.78170 0.78188 0.78206 0.78225 0.78243
+30' +20 0.78261 0.78279 0.78297 0.78315 0.78333 0.78351 0.78369 0.78387 0.78405 0.78424
+40' +10 0.78442 0.78460 0.78478 0.78496 0.78514 0.78532 0.78550 0.78568 0.78586 0.78604
+50' +00 0.78622 0.78640 0.78658 0.78676 0.78694 0.78711 0.78729 0.78747 0.78765 0.78783
=== Sine of 52 degrees / Cosine of 37 degrees ===
+00' +50 0.78801 0.78819 0.78837 0.78855 0.78873 0.78891 0.78908 0.78926 0.78944 0.78962
+10' +40 0.78980 0.78998 0.79016 0.79033 0.79051 0.79069 0.79087 0.79105 0.79122 0.79140
+20' +30 0.79158 0.79176 0.79193 0.79211 0.79229 0.79247 0.79264 0.79282 0.79300 0.79318
+30' +20 0.79335 0.79353 0.79371 0.79388 0.79406 0.79424 0.79441 0.79459 0.79477 0.79494
+40' +10 0.79512 0.79530 0.79547 0.79565 0.79583 0.79600 0.79618 0.79635 0.79653 0.79671
+50' +00 0.79688 0.79706 0.79723 0.79741 0.79758 0.79776 0.79793 0.79811 0.79829 0.79846
=== Sine of 53 degrees / Cosine of 36 degrees ===
+00' +50 0.79864 0.79881 0.79899 0.79916 0.79934 0.79951 0.79968 0.79986 0.80003 0.80021
+10' +40 0.80038 0.80056 0.80073 0.80091 0.80108 0.80125 0.80143 0.80160 0.80178 0.80195
+20' +30 0.80212 0.80230 0.80247 0.80264 0.80282 0.80299 0.80316 0.80334 0.80351 0.80368
+30' +20 0.80386 0.80403 0.80420 0.80438 0.80455 0.80472 0.80489 0.80507 0.80524 0.80541
+40' +10 0.80558 0.80576 0.80593 0.80610 0.80627 0.80644 0.80662 0.80679 0.80696 0.80713
+50' +00 0.80730 0.80748 0.80765 0.80782 0.80799 0.80816 0.80833 0.80850 0.80867 0.80885
=== Sine of 54 degrees / Cosine of 35 degrees ===
+00' +50 0.80902 0.80919 0.80936 0.80953 0.80970 0.80987 0.81004 0.81021 0.81038 0.81055
+10' +40 0.81072 0.81089 0.81106 0.81123 0.81140 0.81157 0.81174 0.81191 0.81208 0.81225
+20' +30 0.81242 0.81259 0.81276 0.81293 0.81310 0.81327 0.81344 0.81361 0.81378 0.81395
+30' +20 0.81412 0.81428 0.81445 0.81462 0.81479 0.81496 0.81513 0.81530 0.81546 0.81563
+40' +10 0.81580 0.81597 0.81614 0.81631 0.81647 0.81664 0.81681 0.81698 0.81714 0.81731
+50' +00 0.81748 0.81765 0.81782 0.81798 0.81815 0.81832 0.81848 0.81865 0.81882 0.81899
=== Sine of 55 degrees / Cosine of 34 degrees ===
+00' +50 0.81915 0.81932 0.81949 0.81965 0.81982 0.81999 0.82015 0.82032 0.82048 0.82065
+10' +40 0.82082 0.82098 0.82115 0.82132 0.82148 0.82165 0.82181 0.82198 0.82214 0.82231
+20' +30 0.82248 0.82264 0.82281 0.82297 0.82314 0.82330 0.82347 0.82363 0.82380 0.82396
+30' +20 0.82413 0.82429 0.82446 0.82462 0.82478 0.82495 0.82511 0.82528 0.82544 0.82561
+40' +10 0.82577 0.82593 0.82610 0.82626 0.82643 0.82659 0.82675 0.82692 0.82708 0.82724
+50' +00 0.82741 0.82757 0.82773 0.82790 0.82806 0.82822 0.82839 0.82855 0.82871 0.82887
=== Sine of 56 degrees / Cosine of 33 degrees ===
+00' +50 0.82904 0.82920 0.82936 0.82953 0.82969 0.82985 0.83001 0.83017 0.83034 0.83050
+10' +40 0.83066 0.83082 0.83098 0.83115 0.83131 0.83147 0.83163 0.83179 0.83195 0.83212
+20' +30 0.83228 0.83244 0.83260 0.83276 0.83292 0.83308 0.83324 0.83340 0.83356 0.83373
+30' +20 0.83389 0.83405 0.83421 0.83437 0.83453 0.83469 0.83485 0.83501 0.83517 0.83533
+40' +10 0.83549 0.83565 0.83581 0.83597 0.83613 0.83629 0.83645 0.83660 0.83676 0.83692
+50' +00 0.83708 0.83724 0.83740 0.83756 0.83772 0.83788 0.83804 0.83819 0.83835 0.83851
=== Sine of 57 degrees / Cosine of 32 degrees ===
+00' +50 0.83867 0.83883 0.83899 0.83915 0.83930 0.83946 0.83962 0.83978 0.83994 0.84009
+10' +40 0.84025 0.84041 0.84057 0.84072 0.84088 0.84104 0.84120 0.84135 0.84151 0.84167
+20' +30 0.84182 0.84198 0.84214 0.84230 0.84245 0.84261 0.84277 0.84292 0.84308 0.84324
+30' +20 0.84339 0.84355 0.84370 0.84386 0.84402 0.84417 0.84433 0.84448 0.84464 0.84480
+40' +10 0.84495 0.84511 0.84526 0.84542 0.84557 0.84573 0.84588 0.84604 0.84619 0.84635
+50' +00 0.84650 0.84666 0.84681 0.84697 0.84712 0.84728 0.84743 0.84759 0.84774 0.84789
=== Sine of 58 degrees / Cosine of 31 degrees ===
+00' +50 0.84805 0.84820 0.84836 0.84851 0.84866 0.84882 0.84897 0.84913 0.84928 0.84943
+10' +40 0.84959 0.84974 0.84989 0.85005 0.85020 0.85035 0.85051 0.85066 0.85081 0.85096
+20' +30 0.85112 0.85127 0.85142 0.85157 0.85173 0.85188 0.85203 0.85218 0.85234 0.85249
+30' +20 0.85264 0.85279 0.85294 0.85310 0.85325 0.85340 0.85355 0.85370 0.85385 0.85401
+40' +10 0.85416 0.85431 0.85446 0.85461 0.85476 0.85491 0.85506 0.85521 0.85536 0.85551
+50' +00 0.85567 0.85582 0.85597 0.85612 0.85627 0.85642 0.85657 0.85672 0.85687 0.85702
=== Sine of 59 degrees / Cosine of 30 degrees ===
+00' +50 0.85717 0.85732 0.85747 0.85762 0.85777 0.85792 0.85806 0.85821 0.85836 0.85851
+10' +40 0.85866 0.85881 0.85896 0.85911 0.85926 0.85941 0.85956 0.85970 0.85985 0.86000
+20' +30 0.86015 0.86030 0.86045 0.86059 0.86074 0.86089 0.86104 0.86119 0.86133 0.86148
+30' +20 0.86163 0.86178 0.86192 0.86207 0.86222 0.86237 0.86251 0.86266 0.86281 0.86295
+40' +10 0.86310 0.86325 0.86340 0.86354 0.86369 0.86384 0.86398 0.86413 0.86427 0.86442
+50' +00 0.86457 0.86471 0.86486 0.86501 0.86515 0.86530 0.86544 0.86559 0.86573 0.86588
=== Sine of 60 degrees / Cosine of 29 degrees ===
+00' +50 0.86603 0.86617 0.86632 0.86646 0.86661 0.86675 0.86690 0.86704 0.86719 0.86733
+10' +40 0.86748 0.86762 0.86777 0.86791 0.86805 0.86820 0.86834 0.86849 0.86863 0.86878
+20' +30 0.86892 0.86906 0.86921 0.86935 0.86949 0.86964 0.86978 0.86993 0.87007 0.87021
+30' +20 0.87036 0.87050 0.87064 0.87079 0.87093 0.87107 0.87121 0.87136 0.87150 0.87164
+40' +10 0.87178 0.87193 0.87207 0.87221 0.87235 0.87250 0.87264 0.87278 0.87292 0.87306
+50' +00 0.87321 0.87335 0.87349 0.87363 0.87377 0.87391 0.87406 0.87420 0.87434 0.87448
=== Sine of 61 degrees / Cosine of 28 degrees ===
+00' +50 0.87462 0.87476 0.87490 0.87504 0.87518 0.87532 0.87546 0.87561 0.87575 0.87589
+10' +40 0.87603 0.87617 0.87631 0.87645 0.87659 0.87673 0.87687 0.87701 0.87715 0.87729
+20' +30 0.87743 0.87756 0.87770 0.87784 0.87798 0.87812 0.87826 0.87840 0.87854 0.87868
+30' +20 0.87882 0.87896 0.87909 0.87923 0.87937 0.87951 0.87965 0.87979 0.87993 0.88006
+40' +10 0.88020 0.88034 0.88048 0.88062 0.88075 0.88089 0.88103 0.88117 0.88130 0.88144
+50' +00 0.88158 0.88172 0.88185 0.88199 0.88213 0.88226 0.88240 0.88254 0.88267 0.88281
=== Sine of 62 degrees / Cosine of 27 degrees ===
+00' +50 0.88295 0.88308 0.88322 0.88336 0.88349 0.88363 0.88377 0.88390 0.88404 0.88417
+10' +40 0.88431 0.88445 0.88458 0.88472 0.88485 0.88499 0.88512 0.88526 0.88539 0.88553
+20' +30 0.88566 0.88580 0.88593 0.88607 0.88620 0.88634 0.88647 0.88661 0.88674 0.88688
+30' +20 0.88701 0.88715 0.88728 0.88741 0.88755 0.88768 0.88782 0.88795 0.88808 0.88822
+40' +10 0.88835 0.88848 0.88862 0.88875 0.88888 0.88902 0.88915 0.88928 0.88942 0.88955
+50' +00 0.88968 0.88981 0.88995 0.89008 0.89021 0.89035 0.89048 0.89061 0.89074 0.89087
=== Sine of 63 degrees / Cosine of 26 degrees ===
+00' +50 0.89101 0.89114 0.89127 0.89140 0.89153 0.89167 0.89180 0.89193 0.89206 0.89219
+10' +40 0.89232 0.89245 0.89259 0.89272 0.89285 0.89298 0.89311 0.89324 0.89337 0.89350
+20' +30 0.89363 0.89376 0.89389 0.89402 0.89415 0.89428 0.89441 0.89454 0.89467 0.89480
+30' +20 0.89493 0.89506 0.89519 0.89532 0.89545 0.89558 0.89571 0.89584 0.89597 0.89610
+40' +10 0.89623 0.89636 0.89649 0.89662 0.89674 0.89687 0.89700 0.89713 0.89726 0.89739
+50' +00 0.89752 0.89764 0.89777 0.89790 0.89803 0.89816 0.89828 0.89841 0.89854 0.89867
=== Sine of 64 degrees / Cosine of 25 degrees ===
+00' +50 0.89879 0.89892 0.89905 0.89918 0.89930 0.89943 0.89956 0.89968 0.89981 0.89994
+10' +40 0.90007 0.90019 0.90032 0.90045 0.90057 0.90070 0.90082 0.90095 0.90108 0.90120
+20' +30 0.90133 0.90146 0.90158 0.90171 0.90183 0.90196 0.90208 0.90221 0.90233 0.90246
+30' +20 0.90259 0.90271 0.90284 0.90296 0.90309 0.90321 0.90334 0.90346 0.90358 0.90371
+40' +10 0.90383 0.90396 0.90408 0.90421 0.90433 0.90446 0.90458 0.90470 0.90483 0.90495
+50' +00 0.90507 0.90520 0.90532 0.90545 0.90557 0.90569 0.90582 0.90594 0.90606 0.90618
=== Sine of 65 degrees / Cosine of 24 degrees ===
+00' +50 0.90631 0.90643 0.90655 0.90668 0.90680 0.90692 0.90704 0.90717 0.90729 0.90741
+10' +40 0.90753 0.90766 0.90778 0.90790 0.90802 0.90814 0.90826 0.90839 0.90851 0.90863
+20' +30 0.90875 0.90887 0.90899 0.90911 0.90924 0.90936 0.90948 0.90960 0.90972 0.90984
+30' +20 0.90996 0.91008 0.91020 0.91032 0.91044 0.91056 0.91068 0.91080 0.91092 0.91104
+40' +10 0.91116 0.91128 0.91140 0.91152 0.91164 0.91176 0.91188 0.91200 0.91212 0.91224
+50' +00 0.91236 0.91248 0.91260 0.91272 0.91283 0.91295 0.91307 0.91319 0.91331 0.91343
=== Sine of 66 degrees / Cosine of 23 degrees ===
+00' +50 0.91355 0.91366 0.91378 0.91390 0.91402 0.91414 0.91425 0.91437 0.91449 0.91461
+10' +40 0.91472 0.91484 0.91496 0.91508 0.91519 0.91531 0.91543 0.91555 0.91566 0.91578
+20' +30 0.91590 0.91601 0.91613 0.91625 0.91636 0.91648 0.91660 0.91671 0.91683 0.91694
+30' +20 0.91706 0.91718 0.91729 0.91741 0.91752 0.91764 0.91775 0.91787 0.91799 0.91810
+40' +10 0.91822 0.91833 0.91845 0.91856 0.91868 0.91879 0.91891 0.91902 0.91914 0.91925
+50' +00 0.91936 0.91948 0.91959 0.91971 0.91982 0.91994 0.92005 0.92016 0.92028 0.92039
=== Sine of 67 degrees / Cosine of 22 degrees ===
+00' +50 0.92050 0.92062 0.92073 0.92085 0.92096 0.92107 0.92119 0.92130 0.92141 0.92152
+10' +40 0.92164 0.92175 0.92186 0.92198 0.92209 0.92220 0.92231 0.92243 0.92254 0.92265
+20' +30 0.92276 0.92287 0.92299 0.92310 0.92321 0.92332 0.92343 0.92355 0.92366 0.92377
+30' +20 0.92388 0.92399 0.92410 0.92421 0.92432 0.92444 0.92455 0.92466 0.92477 0.92488
+40' +10 0.92499 0.92510 0.92521 0.92532 0.92543 0.92554 0.92565 0.92576 0.92587 0.92598
+50' +00 0.92609 0.92620 0.92631 0.92642 0.92653 0.92664 0.92675 0.92686 0.92697 0.92707
=== Sine of 68 degrees / Cosine of 21 degrees ===
+00' +50 0.92718 0.92729 0.92740 0.92751 0.92762 0.92773 0.92784 0.92794 0.92805 0.92816
+10' +40 0.92827 0.92838 0.92849 0.92859 0.92870 0.92881 0.92892 0.92902 0.92913 0.92924
+20' +30 0.92935 0.92945 0.92956 0.92967 0.92978 0.92988 0.92999 0.93010 0.93020 0.93031
+30' +20 0.93042 0.93052 0.93063 0.93074 0.93084 0.93095 0.93106 0.93116 0.93127 0.93137
+40' +10 0.93148 0.93159 0.93169 0.93180 0.93190 0.93201 0.93211 0.93222 0.93232 0.93243
+50' +00 0.93253 0.93264 0.93274 0.93285 0.93295 0.93306 0.93316 0.93327 0.93337 0.93348
=== Sine of 69 degrees / Cosine of 20 degrees ===
+00' +50 0.93358 0.93368 0.93379 0.93389 0.93400 0.93410 0.93420 0.93431 0.93441 0.93452
+10' +40 0.93462 0.93472 0.93483 0.93493 0.93503 0.93514 0.93524 0.93534 0.93544 0.93555
+20' +30 0.93565 0.93575 0.93585 0.93596 0.93606 0.93616 0.93626 0.93637 0.93647 0.93657
+30' +20 0.93667 0.93677 0.93688 0.93698 0.93708 0.93718 0.93728 0.93738 0.93748 0.93759
+40' +10 0.93769 0.93779 0.93789 0.93799 0.93809 0.93819 0.93829 0.93839 0.93849 0.93859
+50' +00 0.93869 0.93879 0.93889 0.93899 0.93909 0.93919 0.93929 0.93939 0.93949 0.93959
=== Sine of 70 degrees / Cosine of 19 degrees ===
+00' +50 0.93969 0.93979 0.93989 0.93999 0.94009 0.94019 0.94029 0.94039 0.94049 0.94058
+10' +40 0.94068 0.94078 0.94088 0.94098 0.94108 0.94118 0.94127 0.94137 0.94147 0.94157
+20' +30 0.94167 0.94176 0.94186 0.94196 0.94206 0.94215 0.94225 0.94235 0.94245 0.94254
+30' +20 0.94264 0.94274 0.94284 0.94293 0.94303 0.94313 0.94322 0.94332 0.94342 0.94351
+40' +10 0.94361 0.94370 0.94380 0.94390 0.94399 0.94409 0.94418 0.94428 0.94438 0.94447
+50' +00 0.94457 0.94466 0.94476 0.94485 0.94495 0.94504 0.94514 0.94523 0.94533 0.94542
=== Sine of 71 degrees / Cosine of 18 degrees ===
+00' +50 0.94552 0.94561 0.94571 0.94580 0.94590 0.94599 0.94609 0.94618 0.94627 0.94637
+10' +40 0.94646 0.94656 0.94665 0.94674 0.94684 0.94693 0.94702 0.94712 0.94721 0.94730
+20' +30 0.94740 0.94749 0.94758 0.94768 0.94777 0.94786 0.94795 0.94805 0.94814 0.94823
+30' +20 0.94832 0.94842 0.94851 0.94860 0.94869 0.94878 0.94888 0.94897 0.94906 0.94915
+40' +10 0.94924 0.94933 0.94943 0.94952 0.94961 0.94970 0.94979 0.94988 0.94997 0.95006
+50' +00 0.95015 0.95024 0.95033 0.95043 0.95052 0.95061 0.95070 0.95079 0.95088 0.95097
=== Sine of 72 degrees / Cosine of 17 degrees ===
+00' +50 0.95106 0.95115 0.95124 0.95133 0.95142 0.95150 0.95159 0.95168 0.95177 0.95186
+10' +40 0.95195 0.95204 0.95213 0.95222 0.95231 0.95240 0.95248 0.95257 0.95266 0.95275
+20' +30 0.95284 0.95293 0.95301 0.95310 0.95319 0.95328 0.95337 0.95345 0.95354 0.95363
+30' +20 0.95372 0.95380 0.95389 0.95398 0.95407 0.95415 0.95424 0.95433 0.95441 0.95450
+40' +10 0.95459 0.95467 0.95476 0.95485 0.95493 0.95502 0.95511 0.95519 0.95528 0.95536
+50' +00 0.95545 0.95554 0.95562 0.95571 0.95579 0.95588 0.95596 0.95605 0.95613 0.95622
=== Sine of 73 degrees / Cosine of 16 degrees ===
+00' +50 0.95630 0.95639 0.95647 0.95656 0.95664 0.95673 0.95681 0.95690 0.95698 0.95707
+10' +40 0.95715 0.95724 0.95732 0.95740 0.95749 0.95757 0.95766 0.95774 0.95782 0.95791
+20' +30 0.95799 0.95807 0.95816 0.95824 0.95832 0.95841 0.95849 0.95857 0.95865 0.95874
+30' +20 0.95882 0.95890 0.95898 0.95907 0.95915 0.95923 0.95931 0.95940 0.95948 0.95956
+40' +10 0.95964 0.95972 0.95981 0.95989 0.95997 0.96005 0.96013 0.96021 0.96029 0.96037
+50' +00 0.96046 0.96054 0.96062 0.96070 0.96078 0.96086 0.96094 0.96102 0.96110 0.96118
=== Sine of 74 degrees / Cosine of 15 degrees ===
+00' +50 0.96126 0.96134 0.96142 0.96150 0.96158 0.96166 0.96174 0.96182 0.96190 0.96198
+10' +40 0.96206 0.96214 0.96222 0.96230 0.96238 0.96246 0.96253 0.96261 0.96269 0.96277
+20' +30 0.96285 0.96293 0.96301 0.96308 0.96316 0.96324 0.96332 0.96340 0.96347 0.96355
+30' +20 0.96363 0.96371 0.96379 0.96386 0.96394 0.96402 0.96410 0.96417 0.96425 0.96433
+40' +10 0.96440 0.96448 0.96456 0.96463 0.96471 0.96479 0.96486 0.96494 0.96502 0.96509
+50' +00 0.96517 0.96524 0.96532 0.96540 0.96547 0.96555 0.96562 0.96570 0.96578 0.96585
=== Sine of 75 degrees / Cosine of 14 degrees ===
+00' +50 0.96593 0.96600 0.96608 0.96615 0.96623 0.96630 0.96638 0.96645 0.96653 0.96660
+10' +40 0.96667 0.96675 0.96682 0.96690 0.96697 0.96705 0.96712 0.96719 0.96727 0.96734
+20' +30 0.96742 0.96749 0.96756 0.96764 0.96771 0.96778 0.96786 0.96793 0.96800 0.96807
+30' +20 0.96815 0.96822 0.96829 0.96837 0.96844 0.96851 0.96858 0.96866 0.96873 0.96880
+40' +10 0.96887 0.96894 0.96902 0.96909 0.96916 0.96923 0.96930 0.96937 0.96945 0.96952
+50' +00 0.96959 0.96966 0.96973 0.96980 0.96987 0.96994 0.97001 0.97008 0.97015 0.97023
=== Sine of 76 degrees / Cosine of 13 degrees ===
+00' +50 0.97030 0.97037 0.97044 0.97051 0.97058 0.97065 0.97072 0.97079 0.97086 0.97093
+10' +40 0.97100 0.97106 0.97113 0.97120 0.97127 0.97134 0.97141 0.97148 0.97155 0.97162
+20' +30 0.97169 0.97176 0.97182 0.97189 0.97196 0.97203 0.97210 0.97217 0.97223 0.97230
+30' +20 0.97237 0.97244 0.97251 0.97257 0.97264 0.97271 0.97278 0.97284 0.97291 0.97298
+40' +10 0.97304 0.97311 0.97318 0.97325 0.97331 0.97338 0.97345 0.97351 0.97358 0.97365
+50' +00 0.97371 0.97378 0.97384 0.97391 0.97398 0.97404 0.97411 0.97417 0.97424 0.97430
=== Sine of 77 degrees / Cosine of 12 degrees ===
+00' +50 0.97437 0.97444 0.97450 0.97457 0.97463 0.97470 0.97476 0.97483 0.97489 0.97496
+10' +40 0.97502 0.97508 0.97515 0.97521 0.97528 0.97534 0.97541 0.97547 0.97553 0.97560
+20' +30 0.97566 0.97573 0.97579 0.97585 0.97592 0.97598 0.97604 0.97611 0.97617 0.97623
+30' +20 0.97630 0.97636 0.97642 0.97648 0.97655 0.97661 0.97667 0.97673 0.97680 0.97686
+40' +10 0.97692 0.97698 0.97705 0.97711 0.97717 0.97723 0.97729 0.97735 0.97742 0.97748
+50' +00 0.97754 0.97760 0.97766 0.97772 0.97778 0.97784 0.97791 0.97797 0.97803 0.97809
=== Sine of 78 degrees / Cosine of 11 degrees ===
+00' +50 0.97815 0.97821 0.97827 0.97833 0.97839 0.97845 0.97851 0.97857 0.97863 0.97869
+10' +40 0.97875 0.97881 0.97887 0.97893 0.97899 0.97905 0.97910 0.97916 0.97922 0.97928
+20' +30 0.97934 0.97940 0.97946 0.97952 0.97958 0.97963 0.97969 0.97975 0.97981 0.97987
+30' +20 0.97992 0.97998 0.98004 0.98010 0.98016 0.98021 0.98027 0.98033 0.98039 0.98044
+40' +10 0.98050 0.98056 0.98061 0.98067 0.98073 0.98079 0.98084 0.98090 0.98096 0.98101
+50' +00 0.98107 0.98112 0.98118 0.98124 0.98129 0.98135 0.98140 0.98146 0.98152 0.98157
=== Sine of 79 degrees / Cosine of 10 degrees ===
+00' +50 0.98163 0.98168 0.98174 0.98179 0.98185 0.98190 0.98196 0.98201 0.98207 0.98212
+10' +40 0.98218 0.98223 0.98229 0.98234 0.98240 0.98245 0.98250 0.98256 0.98261 0.98267
+20' +30 0.98272 0.98277 0.98283 0.98288 0.98294 0.98299 0.98304 0.98310 0.98315 0.98320
+30' +20 0.98325 0.98331 0.98336 0.98341 0.98347 0.98352 0.98357 0.98362 0.98368 0.98373
+40' +10 0.98378 0.98383 0.98389 0.98394 0.98399 0.98404 0.98409 0.98414 0.98420 0.98425
+50' +00 0.98430 0.98435 0.98440 0.98445 0.98450 0.98455 0.98461 0.98466 0.98471 0.98476
=== Sine of 80 degrees / Cosine of 9 degrees ===
+00' +50 0.98481 0.98486 0.98491 0.98496 0.98501 0.98506 0.98511 0.98516 0.98521 0.98526
+10' +40 0.98531 0.98536 0.98541 0.98546 0.98551 0.98556 0.98561 0.98565 0.98570 0.98575
+20' +30 0.98580 0.98585 0.98590 0.98595 0.98600 0.98604 0.98609 0.98614 0.98619 0.98624
+30' +20 0.98629 0.98633 0.98638 0.98643 0.98648 0.98652 0.98657 0.98662 0.98667 0.98671
+40' +10 0.98676 0.98681 0.98686 0.98690 0.98695 0.98700 0.98704 0.98709 0.98714 0.98718
+50' +00 0.98723 0.98728 0.98732 0.98737 0.98741 0.98746 0.98751 0.98755 0.98760 0.98764
=== Sine of 81 degrees / Cosine of 8 degrees ===
+00' +50 0.98769 0.98773 0.98778 0.98782 0.98787 0.98791 0.98796 0.98800 0.98805 0.98809
+10' +40 0.98814 0.98818 0.98823 0.98827 0.98832 0.98836 0.98841 0.98845 0.98849 0.98854
+20' +30 0.98858 0.98863 0.98867 0.98871 0.98876 0.98880 0.98884 0.98889 0.98893 0.98897
+30' +20 0.98902 0.98906 0.98910 0.98914 0.98919 0.98923 0.98927 0.98931 0.98936 0.98940
+40' +10 0.98944 0.98948 0.98953 0.98957 0.98961 0.98965 0.98969 0.98973 0.98978 0.98982
+50' +00 0.98986 0.98990 0.98994 0.98998 0.99002 0.99006 0.99011 0.99015 0.99019 0.99023
=== Sine of 82 degrees / Cosine of 7 degrees ===
+00' +50 0.99027 0.99031 0.99035 0.99039 0.99043 0.99047 0.99051 0.99055 0.99059 0.99063
+10' +40 0.99067 0.99071 0.99075 0.99079 0.99083 0.99087 0.99091 0.99094 0.99098 0.99102
+20' +30 0.99106 0.99110 0.99114 0.99118 0.99122 0.99125 0.99129 0.99133 0.99137 0.99141
+30' +20 0.99144 0.99148 0.99152 0.99156 0.99160 0.99163 0.99167 0.99171 0.99175 0.99178
+40' +10 0.99182 0.99186 0.99189 0.99193 0.99197 0.99200 0.99204 0.99208 0.99211 0.99215
+50' +00 0.99219 0.99222 0.99226 0.99230 0.99233 0.99237 0.99240 0.99244 0.99248 0.99251
=== Sine of 83 degrees / Cosine of 6 degrees ===
+00' +50 0.99255 0.99258 0.99262 0.99265 0.99269 0.99272 0.99276 0.99279 0.99283 0.99286
+10' +40 0.99290 0.99293 0.99297 0.99300 0.99303 0.99307 0.99310 0.99314 0.99317 0.99320
+20' +30 0.99324 0.99327 0.99331 0.99334 0.99337 0.99341 0.99344 0.99347 0.99351 0.99354
+30' +20 0.99357 0.99360 0.99364 0.99367 0.99370 0.99374 0.99377 0.99380 0.99383 0.99386
+40' +10 0.99390 0.99393 0.99396 0.99399 0.99402 0.99406 0.99409 0.99412 0.99415 0.99418
+50' +00 0.99421 0.99424 0.99428 0.99431 0.99434 0.99437 0.99440 0.99443 0.99446 0.99449
=== Sine of 84 degrees / Cosine of 5 degrees ===
+00' +50 0.99452 0.99455 0.99458 0.99461 0.99464 0.99467 0.99470 0.99473 0.99476 0.99479
+10' +40 0.99482 0.99485 0.99488 0.99491 0.99494 0.99497 0.99500 0.99503 0.99506 0.99508
+20' +30 0.99511 0.99514 0.99517 0.99520 0.99523 0.99526 0.99528 0.99531 0.99534 0.99537
+30' +20 0.99540 0.99542 0.99545 0.99548 0.99551 0.99553 0.99556 0.99559 0.99562 0.99564
+40' +10 0.99567 0.99570 0.99572 0.99575 0.99578 0.99580 0.99583 0.99586 0.99588 0.99591
+50' +00 0.99594 0.99596 0.99599 0.99602 0.99604 0.99607 0.99609 0.99612 0.99614 0.99617
=== Sine of 85 degrees / Cosine of 4 degrees ===
+00' +50 0.99619 0.99622 0.99625 0.99627 0.99630 0.99632 0.99635 0.99637 0.99639 0.99642
+10' +40 0.99644 0.99647 0.99649 0.99652 0.99654 0.99657 0.99659 0.99661 0.99664 0.99666
+20' +30 0.99668 0.99671 0.99673 0.99676 0.99678 0.99680 0.99683 0.99685 0.99687 0.99689
+30' +20 0.99692 0.99694 0.99696 0.99699 0.99701 0.99703 0.99705 0.99708 0.99710 0.99712
+40' +10 0.99714 0.99716 0.99719 0.99721 0.99723 0.99725 0.99727 0.99729 0.99731 0.99734
+50' +00 0.99736 0.99738 0.99740 0.99742 0.99744 0.99746 0.99748 0.99750 0.99752 0.99754
=== Sine of 86 degrees / Cosine of 3 degrees ===
+00' +50 0.99756 0.99758 0.99760 0.99762 0.99764 0.99766 0.99768 0.99770 0.99772 0.99774
+10' +40 0.99776 0.99778 0.99780 0.99782 0.99784 0.99786 0.99788 0.99790 0.99792 0.99793
+20' +30 0.99795 0.99797 0.99799 0.99801 0.99803 0.99804 0.99806 0.99808 0.99810 0.99812
+30' +20 0.99813 0.99815 0.99817 0.99819 0.99821 0.99822 0.99824 0.99826 0.99827 0.99829
+40' +10 0.99831 0.99833 0.99834 0.99836 0.99838 0.99839 0.99841 0.99842 0.99844 0.99846
+50' +00 0.99847 0.99849 0.99851 0.99852 0.99854 0.99855 0.99857 0.99858 0.99860 0.99861
=== Sine of 87 degrees / Cosine of 2 degrees ===
+00' +50 0.99863 0.99864 0.99866 0.99867 0.99869 0.99870 0.99872 0.99873 0.99875 0.99876
+10' +40 0.99878 0.99879 0.99881 0.99882 0.99883 0.99885 0.99886 0.99888 0.99889 0.99890
+20' +30 0.99892 0.99893 0.99894 0.99896 0.99897 0.99898 0.99900 0.99901 0.99902 0.99904
+30' +20 0.99905 0.99906 0.99907 0.99909 0.99910 0.99911 0.99912 0.99913 0.99915 0.99916
+40' +10 0.99917 0.99918 0.99919 0.99921 0.99922 0.99923 0.99924 0.99925 0.99926 0.99927
+50' +00 0.99929 0.99930 0.99931 0.99932 0.99933 0.99934 0.99935 0.99936 0.99937 0.99938
=== Sine of 88 degrees / Cosine of 1 degrees ===
+00' +50 0.99939 0.99940 0.99941 0.99942 0.99943 0.99944 0.99945 0.99946 0.99947 0.99948
+10' +40 0.99949 0.99950 0.99951 0.99952 0.99952 0.99953 0.99954 0.99955 0.99956 0.99957
+20' +30 0.99958 0.99959 0.99959 0.99960 0.99961 0.99962 0.99963 0.99963 0.99964 0.99965
+30' +20 0.99966 0.99966 0.99967 0.99968 0.99969 0.99969 0.99970 0.99971 0.99972 0.99972
+40' +10 0.99973 0.99974 0.99974 0.99975 0.99976 0.99976 0.99977 0.99977 0.99978 0.99979
+50' +00 0.99979 0.99980 0.99980 0.99981 0.99982 0.99982 0.99983 0.99983 0.99984 0.99984
=== Sine of 89 degrees / Cosine of 0 degrees ===
+00' +50 0.99985 0.99985 0.99986 0.99986 0.99987 0.99987 0.99988 0.99988 0.99989 0.99989
+10' +40 0.99989 0.99990 0.99990 0.99991 0.99991 0.99991 0.99992 0.99992 0.99993 0.99993
+20' +30 0.99993 0.99994 0.99994 0.99994 0.99995 0.99995 0.99995 0.99995 0.99996 0.99996
+30' +20 0.99996 0.99996 0.99997 0.99997 0.99997 0.99997 0.99998 0.99998 0.99998 0.99998
+40' +10 0.99998 0.99998 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999
+50' +00 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000



#include <math.h>
#include <stdio.h>


int
main()
{
printf ( "Table of sines and cosines in degrees and minutes.\n\n" );
printf( "sin\t\t+00'\t+01'\t+02'\t+03'\t+04'\t+05'\t+06'\t +07'\t+08'\t+09'\n" );
printf( "\tcos\t+10'\t+09'\t+08'\t+07'\t+06'\t+05'\t+04'\t +03'\t+02'\t+01'\n" );

for ( double d=0.; d <89.999; d++ ) {
printf( "\n=== Sine of %.0f degrees / Cosine of %.0f degrees ===", d, 89.-d );

double m=0.;
while ( m < 59.99 ) {
printf( "\n+%02.0f'\t+%02.0f\t", m, 50.-m );
for ( int i=0; i<10; i++, m++ )
printf( "%.5f\t", sin( ( d + m/60. ) / 360 * 2 * 3.141592653589 ) );
}
}
printf( "\n" );
return 0;
}