# Trigonometry

Right-angled triangle is a triangle in which one angle is a right angle, i.e. a 90-degree angle.

Hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. Two other sides are called legs of the triangle .

Let's choose any non-right angle α of the triangle, then

- the sine of this angle is the ratio of the opposite side to the hypotenuse, denoted as sin(α)
- the cosine of this angle is the ratio of the adjacent leg to the hypotenuse, denoted as cos(α)
- the tangent of this angle is the ratio of the opposite leg to the adjacent leg, denoted as tg(α) or tan(α)

There are reciprocals of these functions:

- cosecant: csc(α) = 1/cos(α)
- secant: sec(α) = 1/cos(α)
- cotangent: cot(α) = 1/tg(α)

## constants

The number *π* is a mathematical constant defined as the ratio of a circle's circumference to its diameter.

`π = 3.14159`

The number *e* or Euler's number is a mathematical constant defined as limit of (1 + 1/n)^{n} where n approaches infinity.

`e = 2.71828`

## radians

The radian is the unit for measuring angles. Denoted by the symbol *rad*

An angle of one radian subtended from the center of a unit circle produces an arc with arc length 1.

There is a formula for the correlation of degrees and radians:

`360° = 2 * π rad`

## basic formulas

**sin**^{2}(α) + cos^{2}(α) = 1
tan(α)*ctg(α) = 1
tan(α) = sin(α)/cos(α)=1/ctg(α)
1+tan^{2}(α) = 1/cos^{2}(α) = sec^{2}(α)
ctg(α) = cos(α)/sin(α) = 1/tg(α)
1+ctg^{2}(a) = 1/sin^{2}(α) = cosec^{2}(α)
sec(α)=1/cos(α)
cosec(α)=1/sin(α)

## angle sum

```
sin(α±β) = sin(α)cos(β)±cos(α)sin(β)
cos(α+β) = cos(α)cos(β)-sin(α)sin(β)
cos(α-β) = cos(α)cos(β)+sin(α)sin(β)
tan(α+β) = (tan(α)+tan(β))/(1-tan(α)tan(β))
= (ctg(α)+ctg(β))/(ctg(α)ctg(β)-1)
tan(α-β) = (tan(α)-tan(β))/(1+tan(α)tan(β))
= (ctg(β)-ctg(α))/(ctg(α)ctg(β)+1)
ctg(α+β) = (ctg(α)ctg(β)-1)/(ctg(α)+ctg(β))
= (1-tan(α)tan(β))/(tan(α)+tan(β))
ctg(α-β) = (ctg(α)ctg(β)-1)/(ctg(β)-ctg(α))
= (1+tan(α)tan(β))/(tan(α)-tan(β))
```

## double angle

`sin(2α) = 2sin(α)cos(α) = 2tan(α) / (1+tan`^{2}(α))
cos(2α) = cos^{2}(α) - sin^{2}(α)
= 2cos^{2}(α) - 1
= 1 - 2sin^{2}(α)
= (1 - tan^{2}(α)) / (1+tan^{2}(α))
= (ctg(α) - tan(α)) / (ctg(α) + tan(α))
tan(2α) = 2tan(α) / (1 - tan^{2}(α))

There is the Chebyshev method for finding the nth multiple angle formula.

## signs in quarters

quater | sin | cos | tg | ctg | sec | cosec |
---|---|---|---|---|---|---|

I | + | + | + | + | + | + |

II | + | - | - | - | - | + |

III | - | - | + | + | - | - |

IV | - | + | - | - | + | - |

## parity and periods formulas

arg | sin | cos | tan | ctg | sec | cosec |
---|---|---|---|---|---|---|

-α 2π-α | -sin(α) | cos(α) | -tan(α) | -ctg(α) | sec(α) | -cosec(α) |

0.5π-α 0.5π+α | cos(α) | -sin(α) sin(α) | -ctg(α) ctg(α) | -tan(α) tan(α) | -cosec(α) cosec(α) | sec(α) |

1.5π+α 1.5π-α | -cos(α) | sin(α) -sin(α) | -ctg(α) ctg(α) | -tan(α) tan(α) | cosec(α) -cosec(α) | -sec(α) |

## product to sum formulas

```
2cos(α)cos(β) = cos(α-β) + cos(α+β)
2sin(α)sin(β) = cos(α-β) + cos(α+β)
2sin(α)cos(β) = sin(α+β) + sin(α-β)
2cos(α)sin(β) = sin(α+β) - sin(α-β)
tan(α)tan(β) = ( cos(α-β) - cos(α+β)) / ( cos(α-β) + cos(α+β))
```

## sum to product formulas

```
sin(α) + sin(β) = 2sin((α+β)/2)cos((α-β)/2)
sin(α) - sin(β) = 2sin((α-β)/2)cos((α+β)/2)
cos(α) + cos(β) = 2cos((α+β)/2)cos((α-β)/2)
cos(α) - cos(β) = -2sin((α+β)/2)cos((α-β)/2)
```

## Euler's formula

In following formula *i* is imaginary unit.

**e**^{i*x} = cos(x) + i*sin(x)
sin(x) = (e^{ix} - e^{-ix}) / 2i
cos(x) = (e^{ix} + e^{-ix}) / 2
tan(x) = (e^{-ix} - e^{ix}) / (e^{ix} + e^{-ix})