With `expr.N()`, inverse trigonometric and inverse hyperbolic functions return
their complex principal value when no real value exists. Complex arguments are
also supported:
```js
ce.parse("\\arcsin(2)").N();
// ➔ 1.571 − 1.317i
ce.parse("\\operatorname{arcosh}(0.5)").N();
// ➔ 1.047i
ce.parse("\\operatorname{arsinh}(1+i)").N();
```
Exact arguments remain symbolic with `evaluate()` unless an exact value is
known.
## Trigonometric Functions
| Function | |
| :--------------------- | :---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| `FromPolarCoordinates` | Converts $$ (\operatorname{radius}, \operatorname{angle}) \longrightarrow (x, y)$$ |
| `ToPolarCoordinates` | Converts $$(x, y) \longrightarrow (\operatorname{radius}, \operatorname{angle})$$ |
| `Hypot` | $$\operatorname{Hypot}(x,y) = \sqrt{x^2+y^2}$$ |
| `Haversine` | $$ \operatorname{Haversine}(z) = \sin(\frac{z}{2})^2 $$ The [Haversine function](https://www.wikidata.org/wiki/Q2528380) was important in navigation because it appears in the haversine formula, which is used to reasonably accurately compute distances on an astronomic spheroid given angular positions (e.g., longitude and latitude). |
| `InverseHaversine` | $$\operatorname{InverseHaversine}(z) = 2 \operatorname{Arcsin}(\sqrt{z})$$ |
## Cardinal Sine and Fresnel Functions
| Function | |
| :--------------------- | :---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| `Sinc` | $$\operatorname{sinc}(x) = \frac{\sin x}{x}$$ with $$\operatorname{sinc}(0) = 1$$. The unnormalized [cardinal sine](https://en.wikipedia.org/wiki/Sinc_function) function. |
| `FresnelS` | $$S(x) = \int_0^x \sin\!\left(\tfrac{\pi t^2}{2}\right) dt$$ The [Fresnel S integral](https://en.wikipedia.org/wiki/Fresnel_integral). Odd function with $$S(\infty) = \tfrac{1}{2}$$. |
| `FresnelC` | $$C(x) = \int_0^x \cos\!\left(\tfrac{\pi t^2}{2}\right) dt$$ The [Fresnel C integral](https://en.wikipedia.org/wiki/Fresnel_integral). Odd function with $$C(\infty) = \tfrac{1}{2}$$. |
## Trigonometric Transformations
`TrigExpand`, `TrigToExp` and `TrigReduce` are transformation verbs for
trigonometric and hyperbolic expressions, in the spirit of `Expand` and
`Factor` for polynomials. They preserve **exactness**: an exact input produces
an exact result, and no numeric approximation is introduced unless you request
one with `.N()`.
expr: value
Expands trigonometric and hyperbolic functions of **sums** and **integer
multiples** of angles into products and powers of functions of the individual
angles. This is the trigonometric analog of `Expand`.
Hyperbolic functions are expanded with their own addition formulas, and
`Sec`, `Csc` and `Cot` are handled as reciprocals of the expanded `Cos`, `Sin`
and `Tan`.
```json
["TrigExpand", ["Sin", ["Add", "a", "b"]]]
// ➔ sin(a)cos(b) + cos(a)sin(b)
["TrigExpand", ["Cos", ["Multiply", 2, "x"]]]
// ➔ cos(x)^2 - sin(x)^2
["TrigExpand", ["Sinh", ["Add", "a", "b"]]]
// ➔ sinh(a)cosh(b) + cosh(a)sinh(b)
```
expr: value
Rewrites products and integer powers of trigonometric and hyperbolic functions
as a linear combination of functions of **multiple angles**. This is the
inverse of `TrigExpand`, and the trigonometric analog of `Factor`.
```json
["TrigReduce", ["Power", ["Sin", "x"], 2]]
// ➔ (1 - cos(2x)) / 2
["TrigReduce", ["Multiply", ["Sin", "x"], ["Cos", "x"]]]
// ➔ sin(2x) / 2
```
expr: value
Rewrites trigonometric and hyperbolic functions in terms of the complex
exponential $e^{ix}$, exactly. Useful for manipulating expressions
algebraically or for exposing the underlying exponential structure.
```json
["TrigToExp", ["Sin", "x"]]
// ➔ -(i/2) e^{ix} + (i/2) e^{-ix}
["TrigToExp", ["Cos", "x"]]
// ➔ (e^{ix} + e^{-ix}) / 2
```
## Trigonometric Simplification
The `trigSimplify()` method applies the Fu algorithm to simplify trigonometric
expressions. This systematic approach uses transformation rules to find simpler
equivalent forms.
```javascript
const expr = ce.parse("\\sin^2(x) + \\cos^2(x)");
expr.trigSimplify(); // Returns: 1
const expr2 = ce.parse("2\\sin(x)\\cos(x)");
expr2.trigSimplify(); // Returns: sin(2x)
```
Alternatively, use the `strategy` option with `simplify()`:
```javascript
expr.simplify({ strategy: 'fu' });
```
### Supported Identities
The Fu algorithm recognizes and applies:
- **Pythagorean identities**: `sin²(x) + cos²(x) = 1`, `1 + tan²(x) = sec²(x)`
- **Reciprocal identities**: `sec(x) = 1/cos(x)`, `csc(x) = 1/sin(x)`
- **Double angle formulas**: `sin(2x) = 2sin(x)cos(x)`, `cos(2x) = cos²(x) - sin²(x)`
- **Product-to-sum**: `sin(x)cos(y) = ½[sin(x+y) + sin(x-y)]`
- **Sum-to-product**: `sin(x) + sin(y) = 2sin((x+y)/2)cos((x-y)/2)`
- **Morrie's law**: Products of cosines with doubled angles
Read more about
Simplification