# Vocabulary/odot

 o. y Pi Times

Rank 0 -- operates on individual atoms of y, producing a result of the same shape -- WHY IS THIS IMPORTANT?

Returns (π times y) given any number y .

```   o. 1
3.14159
o. 1r3    NB. π/3 (slightly above 1 rad)
1.0472
o. i.5
0 3.14159 6.28319 9.42478 12.5664
```

### Common uses

1. Represent, in J, common physics expressions involving π

```   sin=: 1&o.   NB. see below: dyadic (o.)
pi=: o. : ([ * [: o. ])
r=: 10

pi 0 1 2
0 3.14159 6.28319
2 pi r
62.8319
```

You can also use J's 'p'-notation to accurately represent expressions involving π

```   1p1          NB. pi
3.14159
3p2          NB. 3 times pi-squared
29.6088
3* (1p1)^2   NB. (equiv)
29.6088
```

```   rfd=: 180 %~ o.   NB. radians from degrees
dfr=: rfd^:_1     NB. degrees from radians

rfd 180
3.14159
dfr 1p1
180
dfr 0.5p1
90
```

### Use These Combinations

Combinations using o. y that have exceptionally good performance include:

 What it does Type; Precisions; Ranks Syntax Variants; Restrictions Benefits; Bug Warnings e^ π y^ ^@o. y handles large values of y

 x o. y Circle Function

Rank 0 0 -- operates on individual atoms of x and y, producing a result of the same shape -- WHY IS THIS IMPORTANT?

Combines the common trigonometric and hyperbolic functions, and their inverses, without the need for reserved words like sin, cos, etc.

```   cop=:       0&o.   NB. sqrt (1-(y^2))
sin=:       1&o.   NB. sine of y
cos=:       2&o.   NB. cosine of y
tan=:       3&o.   NB. tangent of y
coh=:       4&o.   NB. sqrt (1+(y^2))
sinh=:      5&o.   NB. hyperbolic sine of y
cosh=:      6&o.   NB. hyperbolic cosine of y
tanh=:      7&o.   NB. hyperbolic tangent of y
conh=:      8&o.   NB. sqrt -(1+(y^2))
real=:      9&o.   NB. Real part of y
magn=:     10&o.   NB. Magnitude of y
imag=:     11&o.   NB. Imaginary part of y
angle=:    12&o.   NB. Angle of y

arcsin=:   _1&o.   NB. inverse sine
arccos=:   _2&o.   NB. inverse cosine
arctan=:   _3&o.   NB. inverse tangent
cohn=:     _4&o.   NB. sqrt (_1+(y^2))
arcsinh=:  _5&o.   NB. inverse hyperbolic sine
arccosh=:  _6&o.   NB. inverse hyperbolic cosine
arctanh=:  _7&o.   NB. inverse hyperbolic tangent
nconh=:    _8&o.   NB. -sqrt -(1+(y^2))
same=:     _9&o.   NB. y
conj=:    _10&o.   NB. complex conjugate of y
jdot=:    _11&o.   NB. j. y
expj=:    _12&o.   NB. ^ j. y
```

### Common uses

1. To work with trigonometric functions.

2. 9 o. y (real) and 11 o. y (imag) are the best ways to extract the real and imaginary parts of y.

3. To manipulate screen graphics.

4. cop offers occasional convenience in modifying circle functions to work with complementary y

likewise coh for hyperbolic functions

cop leverages the identity: assert 1 -: (*: sin y) + (*: cos y) for all y

```   sin rfd 30
0.5
cop@sin rfd 60
0.5
```

5. Euler's Identity 0 = 1 + e^ i π^

```   1 + expj 1p1
0
```

### Related Primitives

Real/Imag (+. y), Signum (Unit Circle) (* y), Length/Angle (*. y), Magnitude (* y), Imaginary * Complex (j.), Angle * Polar (r.)