Trigonometry

Constants

Symbol	Value
Degrees	\frac{\pi}{180} = 0.017453292519943295769236907\ldots
Pi	\pi \approx 3.14159265358979323\ldots

With expr.N(), inverse trigonometric and inverse hyperbolic functions return their complex principal value when no real value exists. Complex arguments are also supported:

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	Inverse	Hyperbolic	Area Hyperbolic
Sin	Arcsin	Sinh	Arsinh
Cos	Arccos	Cosh	Arcosh
Tan	Arctan Arctan2	Tanh	Artanh
Cot	Arccot	Coth	Arcoth
Sec	Arcsec	Sech	Arsech
Csc	Arccsc	Csch	Arcsch

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 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 function.
FresnelS	S(x) = \int_0^x \sin\!\left(\tfrac{\pi t^2}{2}\right) dt The Fresnel S 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. 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().

TrigExpand(expr: value) -> 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.

["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)

TrigReduce(expr: value) -> 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.

["TrigReduce", ["Power", ["Sin", "x"], 2]]
// ➔ (1 - cos(2x)) / 2

["TrigReduce", ["Multiply", ["Sin", "x"], ["Cos", "x"]]]
// ➔ sin(2x) / 2

TrigToExp(expr: value) -> 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.

["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.

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():

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