２０１７年度前期・微分積分学I・初等超越関数 1
三角関数＆逆三角関数
-2 -1 0 1 2
−π −π/2 0 π/2 π
sin(x) cos(x) csc(x) sec(x)
-2 -1 0 1 2
−π −π/2 0 π/2 π
tan(x) cot(x)
sin(π/2−x) = cos(x), csc(π/2−x) = sec(x) tan(π/2−x) =−cot(x)
−π/2 0 π/2 π
−π/2 -1 0 1 π/2 π
arcsin(x) arccos(x) sin(x) cos(x)
−π
−π/2 0 π/2 π
−3π/2 −π −π/2 0 π/2 π 3π/2 arctan(x)
arccot(x) tan(x) cot(x)
arccos(x) =π/2−arcsin(x) arccot(x) =π/2−arctan(x)
−π
−π/2 0 π/2 π
-2 -1 0 1 π/2 2 3
arccsc(x) arcsec(x) csc(x) sec(x)
sin : R−→[−1,1], 周期2π, cos : R−→[−1,1], 周期2π, tan :R\ {π/2 +nπ} −→R, 周期π, cot : R\ {nπ} −→R, 周期π, sec : R\ {π/2 +nπ} −→R, 周期2π, csc : R\ {nπ} −→R. 周期2π,
cot(x) = 1/tan(x), sec(x) = 1/cos(x), csc(x) = 1/sin(x)
arccsc(x) = arcsin(1/x), arcsec(x) = arccos(1/x),
arcsin: [−1,1]−→[−π/2, π/2] arcsin(sin(x)) =x, x∈[−π/2, π/2], sin(arcsin(x)) =x, x∈[−1,1], arccos: [−1,1]−→[0, π] arccos(cos(x)) =x, x∈[0, π], cos(arccos(x)) =x, x∈[−1,1], arctan:R−→[−π/2, π/2] arctan(tan(x)) = x, x∈[−π/2, π/2], tan(arctan(x)) = x, x∈R, arccot:R−→[0, π] arccot(cot(x)) =x, x∈[−π/2, π/2]\ {0}, cot(arccot(x)) = x, x∈R, arcsec: (−∞,1]∪[1,∞)−→[0,π]\{π/2} arcsec(sec(x)) =x, x∈[0, π]\ {π/2}, sec(arcsec(x)) =x, x̸∈(−1,1), arccsc: (−∞,1]∪[1,∞)−→[−π/2,π/2]\{0} arccsc(csc(x)) =x, x∈[−π/2, π/2]\ {0}, csc(arccsc(x)) =x, x̸∈(−1,1),
[email protected] May, 2017, Version: 1.2
2 ２０１７年度前期・微分積分学I・初等超越関数
双曲線関数＆逆双曲線関数
-5 -4 -3 -2 -1 0 1 2 3 4 5
-3 -2 -1 0 1 2 3
sinh(x) cosh(x) csch(x) sech(x)
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
tanh(x) coth(x)
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
arcsinh(x) arccosh(x) sinh(x) cosh(x)
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
arctanh(x) arccoth(x) tanh(x) coth(x)
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
arccsch(x) arcsech(x) csch(x) sech(x)
cosh(x) = ex+e−x
2 , sinh(x) = ex−e−x
2 ,
tanh(x) = sinh(x)
cosh(x) coth(x) = 1 tanh(x), sech(x) = 1
cosh(x), csch(x) = 1 sinh(x),
cosh : R−→[1,∞), sinh : R−→R,
tanh : R−→(−1,1), coth : R\ {0} −→(−∞,−1)∪(1,∞), sech : R−→(0,1], csch :R\ {0} −→R\ {0}
arccosh : [1,∞)−→R arcsinh :R−→R,
arctanh : (−1,1)−→R, arccoth : (−∞,−1)∪(1,∞)−→R\ {0} arcsech : (0,1]−→[0,∞) arccsch : R\ {0} −→R\ {0}
May, 2017, Version: 1.2 [email protected]
