mech-math 0.3.4

math/tan
===============================================================================

%% Tangent of argument in radians

1. Usage
-------------------------------------------------------------------------------

```mech:disabled
Y := math/tan(X)
```

2. Description
-------------------------------------------------------------------------------

Computes the tangent of each element of `X`. The input `X` is interpreted in radians, not degrees. The result `Y` has the same shape as the input `X`.

3. Input
-------------------------------------------------------------------------------

| Argument | Kind                     | Description                           |
|----------|--------------------------|---------------------------------------|
| `X`      | `float`, `[float]`       | Input angle(s) specified in radians. Can be real or complex. If `X` is complex, tan returns complex results. |

4. Output
-------------------------------------------------------------------------------

| Argument | Kind                     | Description                           |
|----------|--------------------------|---------------------------------------|
| `Y`      | matches input            | Tangent of the input values. For real `X`, `Y` may take any real value (unbounded). For complex `X`, `Y` may have both real and imaginary parts. The shape of `Y` matches the shape of `X`. |

5. Examples  
-------------------------------------------------------------------------------

(a) Find the tangent of a number 

```mech:ex1
y := math/tan(3.14)
```

(b) Find the tangent for a vector of numbers

```mech:ex2
x := [0, 1.57, 3.14]
y := math/tan(x)
```

(c) Find the tangent for a matrix of numbers

```mech:ex3
x := [0, 1; 3 4]
y := math/tan(x)
```

(d) Find the tangent for a matrix of degrees

```mech:ex4
x := [0, 45; 90, 180]
y := math/tan(x * 3.14 / 180)
```

6. Details
-------------------------------------------------------------------------------

The tangent function is another key idea in trigonometry, relating angles to ratios of sides in right triangles.

In a right triangle, the tangent of an angle $$x$$ is defined as the ratio of the length of the opposite side to the adjacent side:

$$ tan(x) = \frac{\text{opposite}}{\text{adjacent}}

In the unit circle, tangent can be understood as the slope of the line through the origin making an angle $$x$$ with the positive x-axis.

For complex numbers, the tangent function is defined using exponentials:

$$ tan(x) = \frac{sin(x)}{cos(x)} = \frac{e^{ix} - e^{-ix}}{i(e^{ix} + e^{-ix})}

This definition extends the tangent function to work beyond real numbers, allowing it to handle complex inputs.