# формула:математика - тригонометрия ◿

## тригонометрия ◿

### Определения:

• a: Length of Opposite
• b: Length of Adjacent
• c: Length of Hypothenuse
• h: Length of Hypothenuse
• alpha: угол α
• beta: угол β
• gamma: угол γ
• x: a/h
• y: b/h
• z: a/b

#### Теоремы Пифагора

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 = Length of Hypothenuse
a = Length of Opposite
b = Length of Adjacent

$${\color{red} {a}} = \sqrt{ {{\color{blue} {c} }^{2}} - {{\color{OliveGreen} {b} }^{2}} }$$
a = Length of Opposite
c = Length of Hypothenuse
b = Length of Adjacent

$${\color{OliveGreen} {b}} = \sqrt{ {{\color{blue} {c} }^{2}} - {{\color{red} {a} }^{2}} }$$
b = Length of Adjacent
c = Length of Hypothenuse
a = Length of Opposite

$${\color{blue} {c}} = \sqrt{ {{\color{red} {a}}^{2}} + {{\color{OliveGreen} {b}}^{2}} }$$
c = Length of Hypothenuse
a = Length of Opposite
b = Length of Adjacent

$${\color{red} {a}} = \sqrt{ {{\color{blue} {c} }^{2}} - {{\color{OliveGreen} {b} }^{2}} }$$
a = Length of Opposite
c = Length of Hypothenuse
b = Length of Adjacent

$${\color{OliveGreen} {b}} = \sqrt{ {{\color{blue} {c} }^{2}} - {{\color{red} {a} }^{2}} }$$
b = Length of Adjacent
c = Length of Hypothenuse
a = Length of Opposite

#### синус

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 = угол α

$${{\color{red} {\alpha} }} = sin^{-1}( \frac{\color{red} {a}}{\color{blue} {c}} )$$
alpha = угол α
a = Length of Opposite
c = Length of Hypothenuse

$${\color{red} {a}} = sin\left( {{\color{red} {\alpha} }} \right) \times {{\color{blue} {c} }}$$
a = Length of Opposite
alpha = угол α
c = Length of Hypothenuse

$${\color{blue} {c}} = \frac{{\color{red} {a} }}{ sin\left( {{\color{red} {\alpha} }} \right) }$$
h = Length of Hypothenuse
a = Length of Opposite
alpha = угол 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 = угол α

$${{\color{red} {\alpha} }} = cos^{-1}( \frac{\color{OliveGreen} {b}}{\color{blue} {c}} )$$
alpha = угол α
b = Length of Adjacent
h = Length of Hypothenuse

$${\color{OliveGreen} {b}} = cos\left( {{\color{red} {\alpha} }} \right) \times {{\color{blue} {c} }}$$
b = Length of Adjacent
alpha = угол α
h = Length of Hypothenuse

$${\color{blue} {c}} = \frac{{\color{OliveGreen} {b} }}{ cos\left( {{\color{red} {\alpha} }} \right) }$$
h = Length of Hypothenuse
b = Length of Adjacent
alpha = угол α

#### касательный

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 = угол α

$${{\color{red} {\alpha} }} = tan^{-1}( \frac{\color{red} {a}}{\color{OliveGreen} {b}} )$$
alpha = угол α
a = Length of Opposite
b = Length of Adjacent

$${\color{red} {a}} = tan\left( {{\color{red} {\alpha} }} \right) \times {{\color{OliveGreen} {b} }}$$
a = Length of Opposite
alpha = угол α
b = Length of Adjacent

$${\color{OliveGreen} {b}} = \frac{{\color{red} {a} }}{ tan\left( {{\color{red} {\alpha} }} \right) }$$
b = Length of Adjacent
a = Length of Opposite
alpha = угол α

#### тригонометрические преобразования

$${\color{red} {\alpha}} + {\color{OliveGreen} {\beta}} + {\color{blue} {\gamma}} = 180$$
alpha = угол α
beta = угол β
gamma = угол γ

$$cos(alpha)^2+sin(alpha)^2=1$$
alpha = угол α
alpha = угол α

$$tan(alpha)=sin(alpha)/cos(alpha)$$
alpha = угол α
alpha = угол α
alpha = угол α

$$cot(alpha)=1/tan(alpha)$$
alpha = угол α
alpha = угол α

$$sin(alpha)=cos(90-alpha)$$
alpha = угол α
alpha = угол α

$$cos(alpha)=sin(90-alpha)$$
alpha = угол α
alpha = угол α

$$tan(alpha)=cot(90-alpha)$$
alpha = угол α
alpha = угол α

$$sin(2*alpha)=2*sin(alpha)*cos(alpha)$$
alpha = угол α
alpha = угол α
alpha = угол α

$$tan(2*alpha)=2*tan(alpha)/(1-tan(alpha)^2)$$
alpha = угол α
alpha = угол α
alpha = угол α

$$sin(3*alpha)=3*sin(alpha)-4*sin(alpha)^3$$
alpha = угол α
alpha = угол α
alpha = угол α

$$cos(alpha)^2=(1/2)+(1/2)*cos(2*alpha)$$
alpha = угол α
alpha = угол α