Version of 2020-03-02

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Grzegorz Jagodziński

Trigonometry

Definitions of trigonometric functions

basic functions
sine (sin) `sin alpha = y/r`   cosine (cos) `cos alpha = x/r`
tangent (tan, tg) `tan alpha = y / x`   cotangent (cot, ctg) `cot alpha = x/y`
secant (sec) `sec alpha = r/x`   cosecant (csc, cosec) `csc alpha = r/y`
secondary functions
versine (vsin, versin) `"vsin" alpha = 1 - cos alpha`   coversine (cvsin, coversin) `"cvsin" alpha = 1 - sin alpha`
vercosine (vcos, vercos) `"vcos" alpha = 1 + cos alpha`   coverscosine (cvcos, covercos) `"cvcos" alpha = 1 + sin alpha`
haversine (havsin, hav, svsin, sversin, sem) `"havsin" alpha = (1 - cos alpha)/2`   hacoversine (hacvsin, scvsin, scoversin, scv) `"hacvsin" alpha = (1 - sin alpha)/2`
havercosine (havcos, svcos, svercos) `"havcos" alpha = (1 + cos alpha)/2`   hacovercosine (hacvcos, scvcos, scovercos) `"hacvcos" alpha = (1 + sin alpha)/2`
exsecant (exsec) `"exsec" alpha = sec alpha - 1`   excosecant (excsc, excosec) `"excsc" alpha = csc alpha - 1`
special functions
chord (crd) `"crd" alpha = sqrt(sin^2 alpha + "vsin"^2 alpha)`   cochord (ccrd, cocrd) `"ccrd" alpha = sqrt(cos^2 alpha + "cvsin"^2 alpha)`
sagitta (sgt) `"sgt" alpha = "vsin" alpha/2`   cosagitta (csgt, cosgt) `"csgt" alpha = "cvsin" alpha/2`

Functions of sum and difference of angles

Start from the bold-marked segment which length is equal to 1. Go on with the triangle to which the unitary segment belongs: express the length of its two other sides with trigonometrical functions. Then do the same with other triangles, using sides of already known length as the base. Finally, find the needed function of sum or difference of angles.

 
`sin` `(alpha + beta) = sin alpha cos beta + cos alpha sin beta`   `sin` `(alpha - beta) = sin alpha cos beta - cos alpha sin beta`
`cos` `(alpha + beta) = cos alpha cos beta - sin alpha sin beta`   `cos` `(alpha - beta) = cos alpha cos beta + sin alpha sin beta`
 
`tan` `(alpha + beta) = (tan alpha + tan beta)/(1 - tan alpha tan beta)`   `tan` `(alpha - beta) = (tan alpha - tan beta)/(1 + tan alpha tan beta)`
`sec` `(alpha + beta) = (sec alpha sec beta)/(1 - tan alpha tan beta)`   `sec` `(alpha - beta) = (sec alpha sec beta)/(1 + tan alpha tan beta)`
 
`cot` `(alpha + beta) = (cot alpha cot beta - 1)/(cot alpha + cot beta)`   `cot` `(alpha - beta) = (cot alpha cot beta + 1)/(cot beta - cot alpha)`
`csc` `(alpha + beta) = (csc alpha csc beta)/(cot alpha + cot beta)`   `csc` `(alpha - beta) = (csc alpha csc beta)/(cot beta - cot alpha)`

The graphics is a manually modified version of the diagrams placed on Wikipedia