Trigonometric function
In mathematics, the trigonometric functions are a set of functions which relate angles to the sides of a right triangle. There are many trigonometric functions, the 3 most common being sine, cosine, tangent, followed by cotangent, secant and cosecant. The last three are called reciprocal trigonometric functions because they act as the reciprocals of other functions. Secant and cosecant are rarely used.
Function | Abbreviation | Relation |
Sine | sin | [math]\sin \theta = \cos \left(\frac{\pi}{2} - \theta \right) \,[/math] |
Cosine | cos | [math]\cos \theta = \sin \left(\frac{\pi}{2} - \theta \right)\,[/math] |
Tangent | tan (or tg) |
[math]\tan \theta = \frac{\sin \theta}{\cos \theta} = \cot \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\cot \theta} \,[/math] |
Cotangent | cot (or ctg) |
[math]\cot \theta = \frac{\cos \theta}{\sin \theta} = \tan \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\tan \theta} \,[/math] |
Secant | sec | [math]\sec \theta = \frac{1}{\cos \theta} = \csc \left(\frac{\pi}{2} - \theta \right) \,[/math] |
Cosecant | csc (or cosec) |
[math]\csc \theta =\frac{1}{\sin \theta} = \sec \left(\frac{\pi}{2} - \theta \right) \,[/math] |
Contents
Definition
The trigonometric functions sometimes are also called circular functions. They are functions of an angle; they are important when studying triangles, among many other applications. Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle (a circle that has a radius of one).
Right triangle definitions
In order to define the trigonometric functions for the angle A, start with a right triangle that contains the angle A:
We use the following names for the sides of the triangle:
- The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h.
- The opposite side is the side opposite to the angle we are interested in, in this case a.
- The adjacent side is the side that is in contact with the angle we are interested in and the right angle, hence its name. In this case the adjacent side is b.
All triangles are taken to exist in Euclidean geometry so that the inside angles of each triangle sum to π radians (or 180°); therefore, for a right triangle the two non-right angles are between zero and π/2 radians. The reader should note that the following definitions, strictly speaking, only define the trigonometric functions for angles in this range. We extend them to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions.
1) The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case
- [math]\sin A = \frac {\textrm{opposite}} {\textrm{hypotenuse}} = \frac {a} {h}.[/math]
Note that this ratio does not depend on the particular right triangle chosen, as long as it contains the angle A, since all those triangles are similar.
The set of zeroes of sine (i.e., the values of [math]x[/math] for which [math]\sin x =0[/math]) is
- [math]\left\{n\pi\big| n\isin\mathbb{Z}\right\}.[/math]
2) The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case
- [math]\cos A = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} = \frac {b} {h}.[/math]
The set of zeroes of cosine is
- [math]\left\{\frac{\pi}{2}+n\pi\bigg| n\isin\mathbb{Z}\right\}.[/math]
3) The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case
- [math]\tan A = \frac {\textrm{opposite}} {\textrm{adjacent}} = \frac {a} {b}.[/math]
The set of zeroes of tangent is
- [math]\left\{n\pi\big| n\isin\mathbb{Z}\right\}.[/math]
The same set of the sine function since
- [math]\tan A = \frac {\sin A}{\cos A}.[/math]
The remaining three functions are best defined using the above three functions.
4) The cosecant csc(A) is the multiplicative inverse of sin(A), i.e. the ratio of the length of the hypotenuse to the length of the opposite side:
- [math]\csc A = \frac {\textrm{hypotenuse}} {\textrm{opposite}} = \frac {h} {a} [/math].
5) The secant sec(A) is the multiplicative inverse of cos(A), i.e. the ratio of the length of the hypotenuse to the length of the adjacent side:
- [math]\sec A = \frac {\textrm{hypotenuse}} {\textrm{adjacent}} = \frac {h} {b} [/math].
6) The cotangent cot(A) is the multiplicative inverse of tan(A), i.e. the ratio of the length of the adjacent side to the length of the opposite side:
- [math]\cot A = \frac {\textrm{adjacent}} {\textrm{opposite}} = \frac {b} {a} [/math].
Definitions by power series
One can define the trigonometric functions also by using power series:
- [math]\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}[/math]
- [math]\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots = \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!}[/math]
and define tangent, cotangent, secant and cosecant using identities, see below.
Identities
Some important identities:
- [math]\tan x = \frac{\sin x}{\cos x}[/math]
- [math]\cot x = \frac{\cos x}{\sin x}[/math]
- [math]\sec x = \frac{1}{\cos x}[/math]
- [math]\csc x = \frac{1}{\sin x}[/math]
- [math]\sin^2 x + \cos^2 x = 1[/math]
- [math]\sin 2x = 2 \sin x \cos x[/math]
- [math]\cos 2x = \cos x \cos x - \sin x \sin x = \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x[/math]
- [math]\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}[/math]
- [math]\sin \left(x \pm y \right)=\sin x \cos y \pm \cos x \sin y[/math]
- [math]\cos \left(x \pm y \right)=\cos x \cos y \mp \sin x \sin y[/math]
- [math]\tan \left ( x \pm y \right ) = \frac{\tan x \pm \tan y}{1 \mp \tan x \tan y}[/math]
Hyperbolic functions
The hyperbolic functions are like the trigonometric functions in that they have very similar properties. They are defined in terms of the exponential function, which is based on the constant e.
- Hyperbolic sine:
- [math]\sinh x = \frac {e^x - e^{-x}} {2} = \frac {e^{2x} - 1} {2e^x} = \frac {1 - e^{-2x}} {2e^{-x}}.[/math]
- Hyperbolic cosine:
- [math]\cosh x = \frac {e^x + e^{-x}} {2} = \frac {e^{2x} + 1} {2e^x} = \frac {1 + e^{-2x}} {2e^{-x}}.[/math]
- Hyperbolic tangent:
- [math]\tanh x = \frac{\sinh x}{\cosh x} = \frac {e^x - e^{-x}} {e^x + e^{-x}} = \frac{e^{2x} - 1} {e^{2x} + 1} = \frac{1 - e^{-2x}} {1 + e^{-2x}}.[/math]
- Hyperbolic cotangent:
- [math]\coth x = \frac{\cosh x}{\sinh x} = \frac {e^x + e^{-x}} {e^x - e^{-x}} = \frac{e^{2x} + 1} {e^{2x} - 1} = \frac{1 + e^{-2x}} {1 - e^{-2x}}, \qquad x \neq 0.[/math]
- Hyperbolic secant:
- [math]\operatorname{sech}\,x = \frac{1}{\cosh x} = \frac {2} {e^x + e^{-x}} = \frac{2e^x} {e^{2x} + 1} = \frac{2e^{-x}} {1 + e^{-2x}}.[/math]
- Hyperbolic cosecant:
- [math]\operatorname{csch}\,x = \frac{1}{\sinh x} = \frac {2} {e^x - e^{-x}} = \frac{2e^x} {e^{2x} - 1} = \frac{2e^{-x}} {1 - e^{-2x}}, \qquad x \neq 0.[/math]
References
- Joseph, George G., The Crest of the Peacock: Non-European Roots of Mathematics, 2nd ed. Penguin Books, London. (2000). ISBN 0-691-00659-8.
- "Madhava of Sangamagramma", MacTutor History of Mathematics Archive. (2002).
- Weisstein, Eric W., "Tangent" from MathWorld, accessed 21 January 2006.
Other pages
Other websites
- nanoSouffle Online Grapher - Extensible features for graphing functions... works in just about every browser, and JavaScript is simply a plus for real-time updates, and not a requirement.
- Sine and cosine function with an implementation in Rexx.
- Trigonomic Functions - An interactive sketch showing the trigonometric functions in terms of the unit circle. (Requires Java.)
- An excellent Flash animation for learning the unit circle