Lösungen

1. Schwingungen

Materialbereitstellung

Remove["Global`*"];
T = 5;
A[1] = 2;
A[2] = 3;
α[1] = π/8;
α[2] = 3 π/6;
ω = 2 π/T;
φ[t_,k_]:= ω * t + α[k];
f[t_,k_]:= A[k] Cos[φ[t,k]];
φ[A1_,φ1_,A2_,φ2_]:= ArcTan[(A1 Sin[φ1]+A2 Sin[φ2])/(A1 Cos[φ1]+A2 Cos[φ2])];
A[A1_,A2_,φ1_,φ2_]:= Sqrt[A1^2+A2^2+2 A1 A2 Cos[φ2-φ1]];
f[A_,φ_]:= A Cos[φ]

a

A[A[1],A[1],φ[t,1],φ[t,2]] //Simplify

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φ[A[1],A[1],φ[t,1],φ[t,2]] //Simplify

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f[A[A[1],A[1],φ[t,1],φ[t,2]], φ[A[1],A[1],φ[t,1],φ[t,2]]] //Simplify

f[A[A[1],A[1],φ[t,1],φ[t,2]], φ[A[1],A[1],φ[t,1],φ[t,2]]]/.t->0  //Simplify

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f[A[A[1],A[1],φ[t,1],φ[t,2]], φ[A[1],A[1],φ[t,1],φ[t,2]]]/.t->Pi/6 //Simplify

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f[A[A[1],A[1],φ[t,1],φ[t,2]], φ[A[1],A[1],φ[t,1],φ[t,2]]]/.t->Pi/4  //Simplify

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f[A[A[1],A[1],φ[t,1],φ[t,2]], φ[A[1],A[1],φ[t,1],φ[t,2]]]/.t->Pi/3 //Simplify

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b

A[A[1],A[2],φ[t,1],φ[t,2]] //Simplify

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φ[A[1],A[2],φ[t,1],φ[t,2]] //Simplify

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f[A[A[1],A[2],φ[t,1],φ[t,2]], φ[A[1],A[2],φ[t,1],φ[t,2]]] //Simplify

f[A[A[1],A[2],φ[t,1],φ[t,2]], φ[A[1],A[2],φ[t,1],φ[t,2]]]/.t->0 //Simplify

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f[A[A[1],A[2],φ[t,1],φ[t,2]], φ[A[1],A[2],φ[t,1],φ[t,2]]]/.t->Pi/6 //Simplify

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f[A[A[1],A[2],φ[t,1],φ[t,2]], φ[A[1],A[2],φ[t,1],φ[t,2]]]/.t->Pi/4 //Simplify

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f[A[A[1],A[2],φ[t,1],φ[t,2]], φ[A[1],A[2],φ[t,1],φ[t,2]]]/.t->Pi/3 //Simplify

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c   Sin[α] = Cos[α - π/2]

α[1] = π/8 - Pi/2;

A[A[1],A[2],φ[t,1],φ[t,2]] //Simplify

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φ[A[1],A[2],φ[t,1],φ[t,2]] //Simplify

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f[A[A[1],A[2],φ[t,1],φ[t,2]], φ[A[1],A[2],φ[t,1],φ[t,2]]] //Simplify

f[A[A[1],A[2],φ[t,1],φ[t,2]], φ[A[1],A[2],φ[t,1],φ[t,2]]]/.t->0 //Simplify

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f[A[A[1],A[2],φ[t,1],φ[t,2]], φ[A[1],A[2],φ[t,1],φ[t,2]]]/.t->Pi/6 //Simplify

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f[A[A[1],A[2],φ[t,1],φ[t,2]], φ[A[1],A[2],φ[t,1],φ[t,2]]]/.t->Pi/4 //Simplify

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f[A[A[1],A[2],φ[t,1],φ[t,2]], φ[A[1],A[2],φ[t,1],φ[t,2]]]/.t->Pi/3 //Simplify

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d   α1 neu definiert

α[1] = -π/8;

A[A[1],A[2],φ[t,1],φ[t,2]] //Simplify

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φ[A[1],A[2],φ[t,1],φ[t,2]] //Simplify

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f[A[A[1],A[2],φ[t,1],φ[t,2]], φ[A[1],A[2],φ[t,1],φ[t,2]]] //Simplify

f[A[A[1],A[2],φ[t,1],φ[t,2]], φ[A[1],A[2],φ[t,1],φ[t,2]]]/.t->0 //Simplify

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f[A[A[1],A[2],φ[t,1],φ[t,2]], φ[A[1],A[2],φ[t,1],φ[t,2]]]/.t->Pi/6 //Simplify

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f[A[A[1],A[2],φ[t,1],φ[t,2]], φ[A[1],A[2],φ[t,1],φ[t,2]]]/.t->Pi/4 //Simplify

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f[A[A[1],A[2],φ[t,1],φ[t,2]], φ[A[1],A[2],φ[t,1],φ[t,2]]]/.t->Pi/3 //Simplify

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2. Komplexe Zahlen

Materialbereitstellung

Remove["Global`*"];
z1 = 2+I;
z2 = 3-2I;
z3 = -4+3I;
z4 = -6-5I;

a

z = z1 + z2

b

z = z2 - z3

c

z= z1 + 2 z2 - 3 z3 + 4 z4

d

z= z1 - 2 z2 + 3 z3 - 4 z4

e

Remove[z];
Solve[3 z1 - 2 z2 + 4 z3 + 6 z == 5 z2 - 3 z, {z}]

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f

Solve[2 (z1 - 3 z2) + 4 (z3 -z ) - 5 (z4 + 2 z - z2) == 8 z1 - 8 z, {z}]

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