# Visual Tangent Without Math Libraries
Canonical public lesson for tangent.
## One picture
Tangent is slope from an angle.
```text
tan(θ) = rise / run
```
On the unit circle:
```text
tan(θ) = sin(θ) / cos(θ)
```
## Simple idea
Sine tells vertical position. Cosine tells horizontal position. Tangent tells how steep the angle is.
## Important warning
When `cos(θ)` gets close to `0`, tangent gets huge.
At `90°` and `270°`, tangent is undefined because the run is zero.
## Code without imports
```python
def tan_from_sin_cos(sin_value, cos_value):
if cos_value == 0:
return None
return sin_value / cos_value
```
## No-library implementation ideas
- Use sine/cosine lookup tables.
- Compute tangent as `sin / cos`.
- Guard against near-zero cosine.
Source task: `85714e74-52ed-41cf-aadc-014fba02568a`