Formulary:Maths - Trigonometry ◿

Trigonometry ◿

Definitions:

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

Pythagorean Theorem

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

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

$${\color{OliveGreen} {b}} = \sqrt{ {{\color{blue} {c} }^{2}} - {{\color{red} {a} }^{2}} }$$
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

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

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

Sine

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 = Angle α

$${{\color{red} {\alpha} }} = sin^{-1}( \frac{\color{red} {a}}{\color{blue} {c}} )$$
alpha = Angle 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 = Angle 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 = Angle α

Cosine

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 = Angle α

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

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

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

Tangent

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 = Angle α

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

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

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

Trigonometric Transformations

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

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

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

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

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

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

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

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

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

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

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