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`

How to make the graphs of trigonometrical functions

A general remark

If we do not care about great accuracy and allow a relative error of 4.5%, we can assume that `pi = 3`. Using graph paper, let’s assume 1 cm (two grids) per unit on the `OY` axis, and 3 cm (6 grids) per `pi` (i.e. the angle of 180°) on the `OX` axis.

Sine

To make a freehand approximate graph of the function `y = sin x`, let’s mark points on the graph with the following coordinates:

`x = 0`, `y = 0` (the origin of the coordinate system),
`x = pi/6`, `y = 1/2` (from the previous point one square to the right, one square up),
`x = pi/2`, `y = 1` (from the previous point two squares to the right, one square up),
`x = 5/6pi`, `y = 1/2` (from the previous point two squares to the right, one square down),
`x = pi`, `y = 0` (from the previous point one square to the right, one square down),
`x = 7/6 pi`, `y = -1/2` (from the previous point one square to the right, one square down),
`x = 3/2 pi`, `y = -1` (from the previous point two squares to the right, one square down),
`x = 11/6 pi`, `y = -1/2` (from the previous point two squares to the right, one square up),
`x = 2 pi`, `y = 0` (from the previous point one square to the right, one square up).

We start from zero and initially move upwards. We move so as not to exceed the extreme values `y = 1` and `y = -1`, each time changing the height by one square. In the horizontal direction, we use the rhythm `1 - 2 - 2 - 1`, then again `1 - 2 - 2 - 1`, repeated as long as needed.

We connect the marked points with a “smooth” line so that its curvature changes smoothly.

If necessary, we can mark the points `x = pi/4`, `y = 0.7` and `x = pi/3`, `y = 0.85`. The ordinate values here are approximated to half a millimeter. In practice, this accuracy is more than enough, but in ordinary applications it is difficult to achieve. In general, these additional points are not needed to create a chart.

Cosine

We create a graph in a similar way, but we start from the point with coordinates `x = 0`, `y = 1`. We mark the next points exactly to the scheme for the sine function, i.e. `x = pi/6`, `y = 1/2`, then `x = pi/2`, `y = 0`, etc.

Tangent

First, we mark the vertical asymptotes intersecting the line `OX` at the points `x = pi/2`, `x = 3/2 pi`, `x = 5/2 pi`, etc. (which differ by `pi`, because it is the period of the tangent function). If necessary, we also mark the asymptotes `x = -pi/2`, `x = -3/2 pi`, `x = -5/2 pi`, etc.

First of all, we use the following points of the function, with exact coordinates:

`x = 0`, `y = 0`,
`x = pi/4`, `y = 1` (the point `x = pi/4` is located exactly in the middle of the grid, between `x = pi/6` and `x = pi/3`),
`x = 3/4 pi`, `y = -1`,
`x = pi`, `y = 0`.

If necessary, we mark analogous points, at every `pi` to the right or left of the chart.

However, this is probably not enough to draw a nice tangentsoid. We can therefore use points with approximate coordinates, taken with considerable accuracy:

`x = pi/3`, `y = 1,75` (the accurate value is `sqrt(3)`, the relative error equals to about `1.04%`, therefore completely unnoticeable in practice),
`x = 5/12 pi`, `y = 3.75` (the accurate value is `2 + sqrt(3)`, the relative error less than `0.5%`; the value of `x = 5/12 pi` is halfway between `x = pi/3` and `x = pi/2`).

We also mark further points in analogous places, resulting from the central symmetry of the tangent function graph (e.g. `x = 7/12 pi`, `y = -3.75`, or `x = 2/3 pi`, `y = -1.75` ).

The plot point for `x = pi/6` is generally not necessary. If there is such a need, we mark it for `y = 0.6`, which is an approximation of the exact value with a relative error below `4%`.

Cotangent

We plot the cotangent function in the same way as the graph of the tangent function:

`x = 0`, the vertical asymptote,
`x = pi/12`, `y = 3.75`,
`x = pi/6`, `y = 1.75`,
`x = pi/4`, `y = 1`,
`x = pi/2`, `y = 0`,
`x = 3/4 pi`, `y = -1`,
`x = 5/6 pi`, `y = -1.75`,
`x = 11/12 pi`, `y = -3.75`,
`x = pi`, the vertical asymptote.

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

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