Formeln: Mathematik - Trigonometrie ◿

Trigonometrie ◿

Definitionen:

  • a: Länge der Gegenkathete
  • b: Länge der Ankathete
  • c: Länge der Hypothenuse
  • h: Länge der Hypothenuse
  • alpha: Winkel α
  • beta: Winkel β
  • gamma: Winkel γ
  • x: a/h
  • y: b/h
  • z: a/b
Layer 1 a b c a b c Layer 1 C A B α

Satz des Pythagoras

In any right triangle, the area of the square whose side is the hypotenuse (c) is equal to the sum of the areas of the squares whose sides are the two legs (a, b).

\( {\color{blue} {c}}^2 = {{\color{red} {a}}^{2} + {{\color{OliveGreen} {b}}^{2} \)

\( {\color{blue} {c}} = \sqrt{ {{\color{red} {a}}^{2} + {{\color{OliveGreen} {b}}^{2} } \)
c = Länge der Hypothenuse
a = Länge der Gegenkathete
b = Länge der Ankathete

\( {\color{red} {a}} = \sqrt{ {{\color{blue} {c} }^{2} - {{\color{OliveGreen} {b} }^{2} } \)
a = Länge der Gegenkathete
c = Länge der Hypothenuse
b = Länge der Ankathete

\( {\color{OliveGreen} {b}} = \sqrt{ {{\color{blue} {c} }^{2} - {{\color{red} {a} }^{2} } \)
b = Länge der Ankathete
c = Länge der Hypothenuse
a = Länge der Gegenkathete

\( {\color{blue} {c}} = \sqrt{ {{\color{red} {a}}^{2} + {{\color{OliveGreen} {b}}^{2} } \)
c = Länge der Hypothenuse
a = Länge der Gegenkathete
b = Länge der Ankathete

\( {\color{red} {a}} = \sqrt{ {{\color{blue} {c} }^{2} - {{\color{OliveGreen} {b} }^{2} } \)
a = Länge der Gegenkathete
c = Länge der Hypothenuse
b = Länge der Ankathete

\( {\color{OliveGreen} {b}} = \sqrt{ {{\color{blue} {c} }^{2} - {{\color{red} {a} }^{2} } \)
b = Länge der Ankathete
c = Länge der Hypothenuse
a = Länge der Gegenkathete


a c Layer 1 C A B α

Sinus

The sine function is a basic triogemetric function. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.

\( \frac{\color{red} {a}}{\color{blue} {c}} = sin\left( {{\color{red} {\alpha} }} \right) \)
x = a/h
alpha = Winkel α

\( {{\color{red} {\alpha} }} = sin^{-1}( \frac{\color{red} {a}}{\color{blue} {c}} ) \)
alpha = Winkel α
a = Länge der Gegenkathete
c = Länge der Hypothenuse

\( {\color{red} {a}} = sin\left( {{\color{red} {\alpha} }} \right) \times {{\color{blue} {c} }} \)
a = Länge der Gegenkathete
alpha = Winkel α
c = Länge der Hypothenuse

\( {\color{blue} {c}} = \frac{{\color{red} {a} }}{ sin\left( {{\color{red} {\alpha} }} \right) } \)
h = Länge der Hypothenuse
a = Länge der Gegenkathete
alpha = Winkel α


b c Layer 1 C A B α

Kosinus

The cosine function is a basic triogemetric function. In a right triangle, cosine gives the ratio of the length of the side adjacent to an angle to the length of the hypotenuse.

\( \frac{\color{OliveGreen} {b}}{\color{blue} {c}} = cos\left( {{\color{red} {\alpha} }} \right) \)
y = b/h
alpha = Winkel α

\( {{\color{red} {\alpha} }} = cos^{-1}( \frac{\color{OliveGreen} {b}}{\color{blue} {c}} ) \)
alpha = Winkel α
b = Länge der Ankathete
h = Länge der Hypothenuse

\( {\color{OliveGreen} {b}} = cos\left( {{\color{red} {\alpha} }} \right) \times {{\color{blue} {c} }} \)
b = Länge der Ankathete
alpha = Winkel α
h = Länge der Hypothenuse

\( {\color{blue} {c}} = \frac{{\color{OliveGreen} {b} }}{ cos\left( {{\color{red} {\alpha} }} \right) } \)
h = Länge der Hypothenuse
b = Länge der Ankathete
alpha = Winkel α


a b Layer 1 C A B α

Tangens

The tangent function is a basic triogemetric function. In a right triangle, tangent function gives the ratio of the length of the side opposite to an angle to the length of the adjacent.

\( \frac{\color{red} {a}}{\color{OliveGreen} {b}} = tan\left( {{\color{red} {\alpha} }} \right) \)
z = a/b
alpha = Winkel α

\( {{\color{red} {\alpha} }} = tan^{-1}( \frac{\color{red} {a}}{\color{OliveGreen} {b}} ) \)
alpha = Winkel α
a = Länge der Gegenkathete
b = Länge der Ankathete

\( {\color{red} {a}} = tan\left( {{\color{red} {\alpha} }} \right) \times {{\color{OliveGreen} {b} }} \)
a = Länge der Gegenkathete
alpha = Winkel α
b = Länge der Ankathete

\( {\color{OliveGreen} {b}} = \frac{{\color{red} {a} }}{ tan\left( {{\color{red} {\alpha} }} \right) } \)
b = Länge der Ankathete
a = Länge der Gegenkathete
alpha = Winkel α


a b c Layer 1 C A B α β γ

Trigonometrische Umformungen

\( {\color{red} {\alpha}} + {\color{OliveGreen} {\beta}} + {\color{blue} {\gamma}} = 180 \)
alpha = Winkel α
beta = Winkel β
gamma = Winkel γ

\( cos(alpha)^2+sin(alpha)^2=1 \)
alpha = Winkel α
alpha = Winkel α

\( tan(alpha)=sin(alpha)/cos(alpha) \)
alpha = Winkel α
alpha = Winkel α
alpha = Winkel α

\( cot(alpha)=1/tan(alpha) \)
alpha = Winkel α
alpha = Winkel α

\( sin(alpha)=cos(90-alpha) \)
alpha = Winkel α
alpha = Winkel α

\( cos(alpha)=sin(90-alpha) \)
alpha = Winkel α
alpha = Winkel α

\( tan(alpha)=cot(90-alpha) \)
alpha = Winkel α
alpha = Winkel α

\( sin(2*alpha)=2*sin(alpha)*cos(alpha) \)
alpha = Winkel α
alpha = Winkel α
alpha = Winkel α

\( tan(2*alpha)=2*tan(alpha)/(1-tan(alpha)^2) \)
alpha = Winkel α
alpha = Winkel α
alpha = Winkel α

\( sin(3*alpha)=3*sin(alpha)-4*sin(alpha)^3 \)
alpha = Winkel α
alpha = Winkel α
alpha = Winkel α

\( cos(alpha)^2=(1/2)+(1/2)*cos(2*alpha) \)
alpha = Winkel α
alpha = Winkel α