brother
Senior Member
I know the easy way, (with the calculator) but just out curiousity does anyone know how to find the 'inverse of the tangent' WITHOUT using the calculator. Basically doing it by hand.
Math, Trig Electrical
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ericsherman37
Senior Member
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- Oregon Coast
Yeah, and the way to do it by hand is the reason we use a calculator. Check this link:
http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/zoo/invtrig.html
http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/zoo/invtrig.html
ericsherman37
Senior Member
- Location
- Oregon Coast
Inverse trig functions don't work quite the same way as inverting an algebraic or quadratic function, because of the nature of a Unit Circle and Trig rules. The link I provided above has a table a little ways down with some examples of the tangent function and corresponding inverse tangent function.
Here's a unit circle picture:
http://www.math.ucsd.edu/~jarmel/math4c/Unit_Circle_Angles.png
Here's a unit circle picture:
http://www.math.ucsd.edu/~jarmel/math4c/Unit_Circle_Angles.png
ericsherman37
Senior Member
- Location
- Oregon Coast
Sorry for being so wordy on this thing.
As a practical example, say you have a right triangle that you want to evaluate. You know the length of its two shorter sides and want to know the angle of the corner between one of the legs and the hypotenuse.
Using the tangent rules for right triangles, you know that tan(angle) = opposite side/adjacent side.
So if the opposite side is length 3 and the adjacent is 4, then:
tan(angle) = 3/4
The inverse tan function is what you would use to balance the equation.
tan^(-1)(tan(angle)) = tan^(-1)(3/4)
angle = tan^(-1)(3/4)
Look at the unit circle and try to figure out where .75 fits in there... I see lots of pi and lots of square roots, but not many regular ol' decimal numbers. Hence the calculator
As a practical example, say you have a right triangle that you want to evaluate. You know the length of its two shorter sides and want to know the angle of the corner between one of the legs and the hypotenuse.
Using the tangent rules for right triangles, you know that tan(angle) = opposite side/adjacent side.
So if the opposite side is length 3 and the adjacent is 4, then:
tan(angle) = 3/4
The inverse tan function is what you would use to balance the equation.
tan^(-1)(tan(angle)) = tan^(-1)(3/4)
angle = tan^(-1)(3/4)
Look at the unit circle and try to figure out where .75 fits in there... I see lots of pi and lots of square roots, but not many regular ol' decimal numbers. Hence the calculator
ericsherman37
Senior Member
- Location
- Oregon Coast
And for kicks, here's some guy that computed the arctan function with a compass and straightedge construction. Geez.
http://www.cs.uwaterloo.ca/journals/JIS/compass.html
http://www.cs.uwaterloo.ca/journals/JIS/compass.html
mivey
Senior Member
Here are two examples of the series:
arctan(0.82) = 0.6868 1765
n = 0, Si = 0.8200 0000, St = 0.8200 0000
n = 1, Si = -0.1837 8933, St = 0.6362 1067
n = 2, Si = 0.0741 4797, St = 0.7103 5864
n = 3, Si = -0.0356 1221, St = 0.6747 4643
n = 4, Si = 0.0186 2439, St = 0.6933 7082
n = 5, Si = -0.0102 4613, St = 0.6831 2469
n = 6, Si = 0.0058 2957, St = 0.6889 5427
n = 7, Si = -0.0033 9716, St = 0.6855 5710
n = 8, Si = 0.0020 1552, St = 0.6875 7262
n = 9, Si = -0.0012 1258, St = 0.6863 6004
n = 10, Si = 0.0007 3769, St = 0.6870 9773
n = 11, Si = -0.0004 5289, St = 0.6866 4484
n = 12, Si = 0.0002 8016, St = 0.6869 2500
n = 13, Si = -0.0001 7443, St = 0.6867 5057
n = 14, Si = 0.0001 0920, St = 0.6868 5977
n = 15, Si = -0.0000 6869, St = 0.6867 9108
n = 16, Si = 0.0000 4339, St = 0.6868 3447
n = 17, Si = -0.0000 2751, St = 0.6868 0696
n = 18, Si = 0.0000 1749, St = 0.6868 2446
n = 19, Si = -0.0000 1116, St = 0.6868 1330
n = 20, Si = 0.0000 0714, St = 0.6868 2044
n = 21, Si = -0.0000 0458, St = 0.6868 1586
n = 22, Si = 0.0000 0294, St = 0.6868 1880
n = 23, Si = -0.0000 0189, St = 0.6868 1691
n = 24, Si = 0.0000 0122, St = 0.6868 1813
n = 25, Si = -0.0000 0079, St = 0.6868 1734
n = 26, Si = 0.0000 0051, St = 0.6868 1785
n = 27, Si = -0.0000 0033, St = 0.6868 1752
n = 28, Si = 0.0000 0021, St = 0.6868 1773
n = 29, Si = -0.0000 0014, St = 0.6868 1759
n = 30, Si = 0.0000 0009, St = 0.6868 1769
n = 31, Si = -0.0000 0006, St = 0.6868 1763
n = 32, Si = 0.0000 0004, St = 0.6868 1766
n = 33, Si = -0.0000 0003, St = 0.6868 1764
n = 34, Si = 0.0000 0002, St = 0.6868 1766
n = 35, Si = -0.0000 0001, St = 0.6868 1765
n = 36, Si = 0.0000 0001, St = 0.6868 1765
n = 37, Si = 0.0000 0000, St = 0.6868 1765
n = 38, Si = 0.0000 0000, St = 0.6868 1765
n = 39, Si = 0.0000 0000, St = 0.6868 1765
n = 40, Si = 0.0000 0000, St = 0.6868 1765
arctan(-0.93) = -0.7491 4462
n = 0, Si = -0.9300 0000, St = -0.9300 0000
n = 1, Si = 0.2681 1900, St = -0.6618 8100
n = 2, Si = -0.1391 3767, St = -0.8010 1867
n = 3, Si = 0.0859 5727, St = -0.7150 6141
n = 4, Si = -0.0578 2345, St = -0.7728 8486
n = 5, Si = 0.0409 1850, St = -0.7319 6636
n = 6, Si = -0.0299 4574, St = -0.7619 1209
n = 7, Si = 0.0224 4672, St = -0.7394 6537
n = 8, Si = -0.0171 3015, St = -0.7565 9552
n = 9, Si = 0.0132 5630, St = -0.7433 3922
n = 10, Si = -0.0103 7344, St = -0.7537 1265
n = 11, Si = 0.0081 9181, St = -0.7455 2084
n = 12, Si = -0.0065 1829, St = -0.7520 3913
n = 13, Si = 0.0052 2006, St = -0.7468 1907
n = 14, Si = -0.0042 0347, St = -0.7510 2253
n = 15, Si = 0.0034 0102, St = -0.7476 2151
n = 16, Si = -0.0027 6327, St = -0.7503 8478
n = 17, Si = 0.0022 5338, St = -0.7481 3139
n = 18, Si = -0.0018 4360, St = -0.7499 7500
n = 19, Si = 0.0015 1276, St = -0.7484 6224
n = 20, Si = -0.0012 4456, St = -0.7497 0680
n = 21, Si = 0.0010 2636, St = -0.7486 8044
n = 22, Si = -0.0008 4824, St = -0.7495 2868
n = 23, Si = 0.0007 0243, St = -0.7488 2626
n = 24, Si = -0.0005 8273, St = -0.7494 0899
n = 25, Si = 0.0004 8424, St = -0.7489 2475
n = 26, Si = -0.0004 0301, St = -0.7493 2776
n = 27, Si = 0.0003 3589, St = -0.7489 9187
n = 28, Si = -0.0002 8032, St = -0.7492 7219
n = 29, Si = 0.0002 3423, St = -0.7490 3796
n = 30, Si = -0.0001 9594, St = -0.7492 3391
n = 31, Si = 0.0001 6409, St = -0.7490 6981
n = 32, Si = -0.0001 3756, St = -0.7492 0737
n = 33, Si = 0.0001 1542, St = -0.7490 9195
n = 34, Si = -0.0000 9693, St = -0.7491 8888
n = 35, Si = 0.0000 8148, St = -0.7491 0741
n = 36, Si = -0.0000 6854, St = -0.7491 7595
n = 37, Si = 0.0000 5770, St = -0.7491 1825
n = 38, Si = -0.0000 4861, St = -0.7491 6685
n = 39, Si = 0.0000 4098, St = -0.7491 2588
n = 40, Si = -0.0000 3456, St = -0.7491 6044
arctan(0.82) = 0.6868 1765
n = 0, Si = 0.8200 0000, St = 0.8200 0000
n = 1, Si = -0.1837 8933, St = 0.6362 1067
n = 2, Si = 0.0741 4797, St = 0.7103 5864
n = 3, Si = -0.0356 1221, St = 0.6747 4643
n = 4, Si = 0.0186 2439, St = 0.6933 7082
n = 5, Si = -0.0102 4613, St = 0.6831 2469
n = 6, Si = 0.0058 2957, St = 0.6889 5427
n = 7, Si = -0.0033 9716, St = 0.6855 5710
n = 8, Si = 0.0020 1552, St = 0.6875 7262
n = 9, Si = -0.0012 1258, St = 0.6863 6004
n = 10, Si = 0.0007 3769, St = 0.6870 9773
n = 11, Si = -0.0004 5289, St = 0.6866 4484
n = 12, Si = 0.0002 8016, St = 0.6869 2500
n = 13, Si = -0.0001 7443, St = 0.6867 5057
n = 14, Si = 0.0001 0920, St = 0.6868 5977
n = 15, Si = -0.0000 6869, St = 0.6867 9108
n = 16, Si = 0.0000 4339, St = 0.6868 3447
n = 17, Si = -0.0000 2751, St = 0.6868 0696
n = 18, Si = 0.0000 1749, St = 0.6868 2446
n = 19, Si = -0.0000 1116, St = 0.6868 1330
n = 20, Si = 0.0000 0714, St = 0.6868 2044
n = 21, Si = -0.0000 0458, St = 0.6868 1586
n = 22, Si = 0.0000 0294, St = 0.6868 1880
n = 23, Si = -0.0000 0189, St = 0.6868 1691
n = 24, Si = 0.0000 0122, St = 0.6868 1813
n = 25, Si = -0.0000 0079, St = 0.6868 1734
n = 26, Si = 0.0000 0051, St = 0.6868 1785
n = 27, Si = -0.0000 0033, St = 0.6868 1752
n = 28, Si = 0.0000 0021, St = 0.6868 1773
n = 29, Si = -0.0000 0014, St = 0.6868 1759
n = 30, Si = 0.0000 0009, St = 0.6868 1769
n = 31, Si = -0.0000 0006, St = 0.6868 1763
n = 32, Si = 0.0000 0004, St = 0.6868 1766
n = 33, Si = -0.0000 0003, St = 0.6868 1764
n = 34, Si = 0.0000 0002, St = 0.6868 1766
n = 35, Si = -0.0000 0001, St = 0.6868 1765
n = 36, Si = 0.0000 0001, St = 0.6868 1765
n = 37, Si = 0.0000 0000, St = 0.6868 1765
n = 38, Si = 0.0000 0000, St = 0.6868 1765
n = 39, Si = 0.0000 0000, St = 0.6868 1765
n = 40, Si = 0.0000 0000, St = 0.6868 1765
arctan(-0.93) = -0.7491 4462
n = 0, Si = -0.9300 0000, St = -0.9300 0000
n = 1, Si = 0.2681 1900, St = -0.6618 8100
n = 2, Si = -0.1391 3767, St = -0.8010 1867
n = 3, Si = 0.0859 5727, St = -0.7150 6141
n = 4, Si = -0.0578 2345, St = -0.7728 8486
n = 5, Si = 0.0409 1850, St = -0.7319 6636
n = 6, Si = -0.0299 4574, St = -0.7619 1209
n = 7, Si = 0.0224 4672, St = -0.7394 6537
n = 8, Si = -0.0171 3015, St = -0.7565 9552
n = 9, Si = 0.0132 5630, St = -0.7433 3922
n = 10, Si = -0.0103 7344, St = -0.7537 1265
n = 11, Si = 0.0081 9181, St = -0.7455 2084
n = 12, Si = -0.0065 1829, St = -0.7520 3913
n = 13, Si = 0.0052 2006, St = -0.7468 1907
n = 14, Si = -0.0042 0347, St = -0.7510 2253
n = 15, Si = 0.0034 0102, St = -0.7476 2151
n = 16, Si = -0.0027 6327, St = -0.7503 8478
n = 17, Si = 0.0022 5338, St = -0.7481 3139
n = 18, Si = -0.0018 4360, St = -0.7499 7500
n = 19, Si = 0.0015 1276, St = -0.7484 6224
n = 20, Si = -0.0012 4456, St = -0.7497 0680
n = 21, Si = 0.0010 2636, St = -0.7486 8044
n = 22, Si = -0.0008 4824, St = -0.7495 2868
n = 23, Si = 0.0007 0243, St = -0.7488 2626
n = 24, Si = -0.0005 8273, St = -0.7494 0899
n = 25, Si = 0.0004 8424, St = -0.7489 2475
n = 26, Si = -0.0004 0301, St = -0.7493 2776
n = 27, Si = 0.0003 3589, St = -0.7489 9187
n = 28, Si = -0.0002 8032, St = -0.7492 7219
n = 29, Si = 0.0002 3423, St = -0.7490 3796
n = 30, Si = -0.0001 9594, St = -0.7492 3391
n = 31, Si = 0.0001 6409, St = -0.7490 6981
n = 32, Si = -0.0001 3756, St = -0.7492 0737
n = 33, Si = 0.0001 1542, St = -0.7490 9195
n = 34, Si = -0.0000 9693, St = -0.7491 8888
n = 35, Si = 0.0000 8148, St = -0.7491 0741
n = 36, Si = -0.0000 6854, St = -0.7491 7595
n = 37, Si = 0.0000 5770, St = -0.7491 1825
n = 38, Si = -0.0000 4861, St = -0.7491 6685
n = 39, Si = 0.0000 4098, St = -0.7491 2588
n = 40, Si = -0.0000 3456, St = -0.7491 6044
Last edited:
iaov
Senior Member
- Location
- Rhinelander WI
I love it when you guys talk dirty like this!
drbond24
Senior Member
- Location
- Barboursville, West Virginia
mivey
Senior Member
I reject your reality and substitute my own.1/tan
He said 'inverse of the tangent' not 'inverse tangent'.
JohnJ0906
Senior Member
- Location
- Baltimore, MD
I reject your reality and substitute my own.
That would be a great signature line. :grin:
mivey
Senior Member
So let be written. So let it be done.
drbond24
Senior Member
- Location
- Barboursville, West Virginia
Ok Pharaoh.
They sell t-shirts with this quote written on them. It is from the goofy red-headed guy on Mythbusters.
mivey
Senior Member
Behold, the house was smitten with frogs, even unto the keyboard, and didst cause mine error.
SegDog
Member
- Location
- Philadelphia
math student
math student
Are you asking for arctan (inverse function) or reciprocal (cotangent)??? This is on the test...
math student
Are you asking for arctan (inverse function) or reciprocal (cotangent)??? This is on the test...
GeorgeB
ElectroHydraulics engineer (retired)
- Location
- Greenville SC
- Occupation
- Retired
One piece of confusion is that "the angle whose tangent is___" has multiple ways to be expressed.
tan(-1 exponent)
atan
arctan (and of course arc tangent)
while the -1 exponent usually is the same thing as reciprocal, with trig function "angle whose (function value) is___" that is normally not the reciprocal.
LarryFine
Master Electrician Electric Contractor Richmond VA
- Location
- Henrico County, VA
- Occupation
- Electrical Contractor
Now, there's a ribbeting story.Behold, the house was smitten with frogs, even unto the keyboard, and didst cause mine error.
SegDog
Member
- Location
- Philadelphia
pity
pity
I am tutoring a guy that's at least 30 years younger. I'm teaching him to write things out so that he really understands the importance of visuals, language and logic.
He keeps telling me that he was trained to use the calculator only. I never appreciated what was meant by the "paperless society" until now. The piles of paper that I have didn't so it...
This week "even-odd trig functions"...wish me luck!
pity
I am tutoring a guy that's at least 30 years younger. I'm teaching him to write things out so that he really understands the importance of visuals, language and logic.
He keeps telling me that he was trained to use the calculator only. I never appreciated what was meant by the "paperless society" until now. The piles of paper that I have didn't so it...
This week "even-odd trig functions"...wish me luck!
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