# Secret trigonometry

## Hidden Trigonometry functions

The versine or versed sine is a trigonometric function found in some of the  earliest (Vedic Aryabhatia I) trigonometric tables.  
The versine of an angle is 1 minus its cosine.  
There are several related functions, most notably the coversine and haversine.  
The latter, half a versine, is of particular importance in
the haversine formula of navigation  
(source: Wikipedia)

### vsin(Θ) = 1 - cos(Θ)

![vsin](./assets/vsin.png)

### cvsin(Θ) = 1 - sin(Θ)

![cvsin](./assets/cvsin.png)

### vcos(Θ) = 1 + cos(Θ)

![vcos](./assets/vcos.png)

### cvcos(Θ) = 1 + sin(Θ)

![cvcos](./assets/cvcos.png)

### hvsin(Θ) = (1 - cos(Θ)) / 2

![hvsin](./assets/hvsin.png)

### hcvsin(Θ) = (1 - sin(Θ)) / 2

![hcvsin](./assets/hcvsin.png)

### hvcos(Θ) = (1 + cos(Θ)) / 2

![hvcos](./assets/hvcos.png)

### hcvcos(Θ) = (1 + sin(Θ)) / 2

![hcvcos](./assets/hcvcos.png)

## Hidden Trigonometry calculus, derivates and integrals

### Examples

```txt
derivates
---------
d(vsin x) / dx = sin x
d(cvsin x) / dx = -cos x

integrals
---------
∫ vsin(x) dx = x - sin x + C
∫ cvsin(x) dx = x + cos x + C
```