Matheseiten-Überblick
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Trigonometrische Zusammenhänge, Vereinfachungen, Formeln

sin(x) = sqrt(1-cos(x)^2) = tan(x)/sqrt(1+tan(x)^2) = 1/sqrt(1+cot(x)^2)
cos(x) = sqrt(1- sin(x)^2) = 1/sqrt(1+tan(x)^2) = cot(x)/sqrt(1+cot(x)^2)
tan(x) = sin(x)/sqrt(1-sin(x)^2) = sqrt(1-cos(x)^2)/cos(x) = 1/cot(x)
cot(x) = sqrt(1-sin(x)^2)/sin(x) = cos(x)/sqrt(1-cos(x)^2) = 1/tan(x)
sin(x)^2 + cos(x)^2 = 1
sec(x)^2 - tan(x)^2 = 1
csc(x)^2 - cot(x)^2 = 1
sin(x)*csc(x) = 1
cos(x)*sec(x) = 1
tan(x)*cot(x) = 1
tan(x) = sin(x)/cos(x)
cot(x) = cos(x)/sin(x)

sin(x+y) = sin(x)*cos(y) + cos(x)*sin(y)
sin(x-y) = sin(x)*cos(y) - cos(x)*sin(y)
cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y)
cos(x-y) = cos(x)*cos(y) + sin(x)*sin(y)
tan(x+y) = (tan(x)+tan(y))/(1-tan(x)*tan(y))
tan(x-y) = (tan(x)-tan(y))/(1+tan(x)*tan(y))
cot(x+y) = (cot(x)*cot(y) - 1)/(cot(y)+cot(x))
cot(x-y) = (cot(x)*cot(y) + 1)/(cot(y)-cot(x))

sin(x+y+z) = sin(x)*cos(y)*cos(z) + cos(x)*sin(y)*cos(z) + cos(x)*cos(y)*sin(z) - sin(x)*sin(y)*sin(z)
cos(x+y+z) = cos(x)*cos(y)*cos(z) - sin(x)*sin(y)*cos(z) - sin(x)*cos(y)*sin(z) - cos(x)*sin(y)*sin(z)

sin(2x) = 2*cos(x)*sin(x)
cos(2x) = cos(x)^2 - sin(x)^2
sin(3x) = 3*sin(x) - 4*sin(x)^3
cos(3x) = 4*cos(x)^3 - 3*cos(x)
sin(4x) = 8*cos(x)^3*sin(x) - 4*cos(x)*sin(x)
cos(4x) = 8*cos(x)^4 - 8*cos(x)^2 + 1
sin(nx) = n*cos(x)^(n-1)*sin(x) - BK(n,3)*cos(x)^(n-3)*sin(x)^3 + BK(n,5)*cos(x)^(n-5)*sin(x)^5 - ...
cos(nx) = cos(x)^n - BK(n,2)*cos(x)^(n-2)*sin(x)^2 + BK(n,4)*cos(x)^(n-4)*sin(x)^4 - ...
sin(·x) = 
cos(·x) = 
      Optionen: Cos(x) eliminieren  Sin(x) eliminieren  automatisch  nach Regel belassen    einzelne Potenzen vollständig auflösen
tan(2x) = 2*tan(x)/(1-tan(x)^2)
cot(2x) = (cot(x)^2-1)/(2*cot(x))
tan(3x) = (3*tan(x) - tan(x)^3)/(1-3*tan(x)^2)
cot(3x) = (cot(x)^3-3*cot(x))/(3*cot(x)^2-1)
tan(4x) = (4*tan(x)-4*tan(x)^3)/(1-6*tan(x)^2+tan(x)^4)
cot(4x) = (cot(x)^4-6*cot(x)^2+1)/(4*cot(x)^3-4*cot(x))

sin(x/2) = sqrt((1-cos(x))/2)
cos(x/2) = sqrt((1+cos(x))/2)
tan(x/2) = sqrt((1-cos(x))/(1+cos(x)) = (1-cos(x))/sin(x) = sin(x)/(1+cos(x))
cot(x/2) = sqrt((1+cos(x))/(1-cos(x)) = (1+cos(x))/sin(x) = sin(x)/(1-cos(x))

sin(x)+sin(y) = 2*sin((x+y)/2)*cos((x-y)/2)
sin(x)-sin(y) = 2*cos((x+y)/2)*sin((x-y)/2)
cos(x)+cos(y) = 2*cos((x+y)/2)*cos((x-y)/2)
cos(x)-cos(y) = -2*sin((x+y)/2)*sin((x-y)/2)
tan(x)+tan(y) = sin(x+y)/(cos(x)*cos(y))
tan(x)-tan(y) = sin(x-y)/(cos(x)*cos(y))
cot(x)+cot(y) = sin(x+y)/(sin(x)*sin(y))
cot(x)-cot(y) = sin(x-y)/(sin(x)*sin(y))

sin(x)*sin(y) = (cos(x-y)-cos(x+y))/2
cos(x)*cos(y) = (cos(x-y)+cos(x+y))/2
sin(x)*cos(y) = (sin(x-y)+sin(x+y))/2
sin(x)*sin(y)*sin(z) = (sin(x+y-z)+sin(-x+y+z)+sin(x-y+z)-sin(x+y+z))/4
sin(x)*sin(y)*cos(z) = (-cos(x+y-z)+cos(-x+y+z)+cos(x-y+z)-cos(x+y+z))/4
sin(x)*cos(y)*cos(z) = (sin(x+y-z)-sin(-x+y+z)+sin(x-y+z)+sin(x+y+z))/4
cos(x)*cos(y)*cos(z) = (cos(x+y-z)+cos(-x+y+z)+cos(x-y+z)+cos(x+y+z))/4
sin(x)^2 = (1 - cos(2*x))/2
sin(x)^3 = (3*sin(x) - sin(3*x))/4
sin(x)^4 = (cos(4*x) - 4*cos(2*x) + 3)/8
cos(x)^2 = (1 + cos(2*x))/2
cos(x)^3 = (cos(3*x) + 3*cos(x))/4
cos(x)^4 = (cos(4*x) + 4*cos(2*x) + 3)/8
sin(x)^2*cos(x) = cos(x) - cos(x)^3
sin(x)*cos(x)^2 = sin(x) - sin(x)^3
sin(x)^2*cos(x)^2 = cos(x)^2 - cos(x)^4

Wird bald fortgesetzt


© Arndt Brünner 1. 11. 2006
Version: 1. 11. 2006