Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and are widely used for studying periodic phenomena through Fourier analysis.

The trigonometric functions most commonly used in modern mathematics are the sine, the cosine, and the tangent functions. Their reciprocals are respectively the cosecant, the secant, and the cotangent functions, which are less commonly used. Each of these six trigonometric functions has a corresponding inverse function and has an analog among the hyperbolic functions. The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine and cosine functions to functions whose domain is the whole real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane with some isolated points removed.

Notation Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are "sin" for sine, "cos" for cosine, "tan" or "tg" for tangent, "sec" for secant, "csc" or "cosec" for cosecant, and "cot" or "ctg" for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as the function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation, for example sin(x). Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression sin ⁡ x + y {\displaystyle \sin x+y} would typically be interpreted to mean ( sin ⁡ x ) + y , {\displaystyle (\sin x)+y,} so parentheses are required to express sin ⁡ ( x + y ) . {\displaystyle \sin(x+y).}

A positive integer appearing as a superscript after the symbol of the function denotes exponentiation, not function composition. For example sin 2 ⁡ x {\displaystyle \sin ^{2}x} and sin 2 ⁡ ( x ) {\displaystyle \sin ^{2}(x)} denote ( sin ⁡ x ) 2 , {\displaystyle (\sin x)^{2},} not sin ⁡ ( sin ⁡ x ) . {\displaystyle \sin(\sin x).} This differs from the (historically later) general functional notation in which f 2 ( x ) = ( f ∘ f ) ( x ) = f ( f ( x ) ) . {\displaystyle f^{2}(x)=(f\circ f)(x)=f(f(x)).}

In contrast, the superscript − 1 {\displaystyle -1} is commonly used to denote the inverse function, not the reciprocal. For example sin − 1 ⁡ x {\displaystyle \sin ^{-1}x} and sin − 1 ⁡ ( x ) {\displaystyle \sin ^{-1}(x)} denote the inverse trigonometric function alternatively written arcsin ⁡ x . {\displaystyle \arcsin x\,.} The equation θ = sin − 1 ⁡ x {\displaystyle \theta =\sin ^{-1}x} implies sin ⁡ θ = x , {\displaystyle \sin \theta =x,} not θ ⋅ sin ⁡ x = 1. {\displaystyle \theta \cdot \sin x=1.} In this case, the superscript could be considered as denoting a composed or iterated function, but negative superscripts other than − 1 {\displaystyle {-1}} are not in common use.

Right-angled triangle definitions

If the acute angle θ is given, then any right triangles that have an angle of θ are similar to each other. This means that the ratio of any two side lengths depends only on θ. Thus these six ratios define six functions of θ, which are the trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, opposite represents the side opposite the given angle θ, and adjacent represents the side between the angle θ and the right angle.

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