2019-수학-가형-홀수-20¶
문제¶
풀이¶
from sympy import *
an, x, y = symbols('an x y')
점 \((a_n, \sin a_n)\)에서의 접선의 방정식은 다음과 같다:
slope = diff(sin(x), x).subs(x,an)
tangent_eqn = slope*(x-an) +sin(an) - y
tangent_eqn
\[\displaystyle - y + \left(- an + x\right) \cos{\left(an \right)} + \sin{\left(an \right)}\]
접선의 방정식이 점 \((-\pi/2,0)\)을 지나므로, \(a_n\)은 다음 관계식을 만족한다:
expr = tangent_eqn.subs({x:-pi/2, y:0})
expr
\[\displaystyle \left(- an - \frac{\pi}{2}\right) \cos{\left(an \right)} + \sin{\left(an \right)}\]
각 자연수 \(n\)에 대하여, \(a_n\)은 \((-a_n - \pi/2)\cos(a_n) + \sin(a_n)=0, \, a_n\in ((n-1)\pi, n\pi)\)을 만족시킨다.
def a_val(n):
#구간 ((n−1)*Pi,n*Pi)의 중심인 (2*n-1)*pi/2) 근방의 수치해를 구한다
return nsolve(expr, an, N((2*n-1)*pi/2))
for n in range(1,20):
print(N((2*n-1)*pi/2)-a_val(n))
0.343206607805906
0.161934840281518
0.106911498978314
0.0799162191353915
0.0638348976159389
0.0531515553286042
0.0455356951475103
0.0398308166062478
0.0353973070478659
0.0318525180080762
0.0289534340256594
0.0265382779645478
0.0244951700327363
0.0227442613616091
0.0212270334845499
0.0198996198622510
0.0187284893008552
0.0176875707976407
0.0167562877655456
\(a_n = \frac{(2n-1)\pi}{2}-\epsilon_n\)라 쓰면, \(\epsilon_n>0\)은 0으로 수렴하는 감소수열임을 관찰할 수 있다.
ㄱ.
for n in range(1,30):
print(tan(a_val(n))-a_val(n)-N(pi/2))
8.88178419700125e-16
9.76996261670138e-15
4.44089209850063e-15
-6.03961325396085e-14
-2.07833750209829e-13
-2.82440737464640e-13
-6.02184968556685e-13
1.03206332369155e-12
8.22453216642316e-13
-1.66799907219684e-12
2.84394729987980e-12
2.48157050464215e-12
-4.79438710954128e-12
2.60236276972137e-12
-3.24540394558426e-12
6.91535717578518e-12
3.13526982154144e-12
6.12843109593086e-13
2.19735341033811e-12
7.87458986906131e-12
1.44897427389878e-11
-2.39079867014880e-11
3.06332736954573e-11
-2.13784545621820e-11
3.41859873742578e-11
-2.12079243055996e-11
7.62057084102707e-13
2.27178276190898e-11
1.46886947050007e-11
ㄴ.
for n in range(1,30):
print(tan(a_val(n+2))-tan(a_val(n))-N(2*pi))
0.236295108827596
0.0820186211460561
0.0430766013621628
0.0267646638065671
0.0182992024680360
0.0133207387236709
0.0101383881010690
0.00797829859547150
0.00644387302422800
0.00531424004767445
0.00445826398528482
0.00379401660305945
0.00326813654973535
0.00284464150367825
0.00249854419007534
0.00221204905830774
0.00197220153437172
0.00176938798246340
0.00159635178234652
0.00144753179616686
0.00131860814368423
0.00120618337671630
0.00110755606861090
0.00102055613471208
0.000943424163075690
0.000874721688028046
0.000813262923465174
0.000758063787365870
0.000708302019098994
ㄷ.
for n in range(1,30):
print(a_val(n+1)+a_val(n+2)-a_val(n)-a_val(n+3))
0.154276487681466
0.0389420197837520
0.0163119375555851
0.00846546133835702
0.00497846374607391
0.00318235062271199
0.00216008950146929
0.00153442557596151
0.00112963297868163
0.000855976050601726
0.000664247389984496
0.000525880054745187
0.000423495048821110
0.000346097315663485
0.000286495119084407
0.000239847529293513
0.000202813560107984
0.000173036205140420
0.000148819942097589
0.000128923700401629
0.000112424753368146
9.86273091285739e-5
8.69999305166402e-5
7.71319380419300e-5
6.87025523973261e-5
6.14587345637574e-5
5.51990580959227e-5
4.97618269719169e-5
4.50162210654526e-5