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Formulary: Maths - Trigonometry ◿

Trigonometry ◿

Definitions:

  • a: Length of Opposite
  • b: Length of Adjacent
  • 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
Layer 1abcc²b²a²abcLayer 1CABα

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
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

acLayer 1CABα

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

bcLayer 1CABα

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
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 = Angle 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 = Angle α

abLayer 1CABα

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
b = Length of Adjacent

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

abcLayer 1CABαβγ

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