> For the complete documentation index, see [llms.txt](https://tlcfem.gitbook.io/suanpan-manual/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://tlcfem.gitbook.io/suanpan-manual/amplitude/universal/trig.md).

# Trig

The amplitude is computed based on the following summation.

For `Sine`,

$$
A(t)=\sum{}a\_i\sin(i\dfrac{2\pi}{T\_0}(t-t\_0)=\sum{}a\_i\sin(if\_0(t-t\_0))\quad\text{for}\quad{}t>t\_0.
$$

For `Cosine`,

$$
A(t)=\sum{}a\_i\cos(i\dfrac{2\pi}{T\_0}(t-t\_0))=\sum{}a\_i\cos(if\_0(t-t\_0))\quad\text{for}\quad{}t>t\_0.
$$

In above equations, $$T\_0$$ is the base period and $$f\_0$$ is base frequency accordingly. In the above definition, $$t\_0$$ is the (pseudo) start time of the step in which the amplitude is defined.

## Syntax

```
amplitude Sine (1) (2) (3) [(4)...]
amplitude Cosine (1) (2) (3) [(4)...]
# (1) int, unique tag
# (2) double, base period T_0
# (3) double, amplitude at base period/frequency a_0
# [(4)...] double, optional amplitudes at higher periods a_i
```

## Example

```
amplitude Sine 1 10. 2.
```

$$
A(t)=2\sin(\dfrac{\pi}{5}(t-t\_0)).
$$

```
amplitude Sine 1 10. 2. 4.
```

$$
A(t)=2\sin(0.2\pi{}(t-t\_0))+4\sin(0.4\pi{}(t-t\_0)).
$$


---

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