# `○` — Cis / Circular
Function. Keys: `Alt-o`. Ranks: `0` monadic, `0 0` dyadic
`○Y`: eiY = cos Y + i sin Y. Real Y is an angle in radians.
Pervasive.
``` bpl
○0 ⍝ 1
○0j1 ⍝ *¯1
```
`K○Y` applies the circular function with code `K`, from 0 to 8. `K○⁻¹Y`
applies its inverse. Pervasive; angles in radians.
``` bpl
1○0 ⍝ 0
2○0 ⍝ 1
1○⁻¹1 ⍝ 1.5707963267948966
```
k |
k○y |
k○⁻¹y |
| 0 |
√(1−y²) |
√(1−y²) |
| 1 |
sin |
arcsin |
| 2 |
cos |
arccos |
| 3 |
tan |
arctan |
| 4 |
√(1+y²) |
APL branch of √(y²−1) |
| 5 |
sinh |
arcsinh |
| 6 |
cosh |
arccosh |
| 7 |
tanh |
arctanh |
| 8 |
√(-1−y²) |
-√(-1−y²) |
Uses complex continuation where needed. The cuts of `3○⁻¹` and `5○⁻¹`
lie on the imaginary axis. As in C99 and NumPy, the sign of a zero real
part chooses the side of these cuts: `3○⁻¹ 0j2` and `3○⁻¹ ¯0j2` give
results on opposite sides. On a cut along the real axis, results always
come from the side above the cut, because BPL turns a number whose
imaginary part is zero, `¯0` included, into a real. For example, `2j¯0`
is `2`. A real argument computes with an imaginary part of +0:
``` bpl
1○⁻¹2 ⍝ 1.5707963267948966j1.3169578969248166
```
APL difference: there are no negative codes or codes 9 to 12. Write
`¯k○Y` as `k○⁻¹Y`. `∨Y` gives the real and imaginary parts, `|Y` the
magnitude, and `∧Y` the magnitude and phase. On a branch cut, APL takes
the side below the cut for a positive real argument.
## Inverse
`○⁻¹Y` gives the angle of the unit complex number `Y`. The table above
gives the inverse of each `K○`.
``` bpl
○⁻¹0j1 ⍝ π0.5
```