Trigonometric functions

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All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.
All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.
Trigonometric functions: Sine, Cosine, Tangent, Cosecant, Secant, Cotangent
Trigonometric functions: Sine, Cosine, Tangent, Cosecant, Secant, Cotangent

In mathematics there are 6 trigonometric functions: sine, cosine, tangent, cotangent, secant and cosecant. Secant and cosecant are rarely used.

Function Abbreviation Relation
Sine sin \sin \theta = \cos \left(\frac{\pi}{2} - \theta \right) \,
Cosine cos \cos \theta = \sin \left(\frac{\pi}{2} - \theta \right)\,
Tangent tan
(or tg)
\tan \theta = \frac{\sin \theta}{\cos \theta} = \cot \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\cot \theta} \,
Cotangent cot
(or ctg)
\cot \theta = \frac{\cos \theta}{\sin \theta} = \tan \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\tan \theta} \,
Secant sec \sec \theta = \frac{1}{\cos \theta} = \csc \left(\frac{\pi}{2} - \theta \right) \,
Cosecant csc
(or cosec)
\csc \theta =\frac{1}{\sin \theta} = \sec \left(\frac{\pi}{2} - \theta \right) \,

Contents

[edit] Definition

The trigonometric functions sometimes are also called circular functions. They are functions of an angle; they are important when studying triangles and modeling periodic phenomena, 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 or a circle that has the radius of one.

[edit] Right triangle definitions

A right triangle always includes a 90° (π/2 radians) angle, here labeled C.  Angles A and B may vary.  Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle.
A right triangle always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle.

In order to define the trigonometric functions for the angle A, start with an 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 the Euclidean plane 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

\sin A = \frac {\textrm{opposite}} {\textrm{hypotenuse}} = \frac {a} {h}.

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 x for which sinx = 0) is

\left\{n\pi\big| n\isin\mathbb{Z}\right\}.

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

\cos A = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} = \frac {b} {h}.

The set of zeroes of cosine is

\left\{\frac{\pi}{2}+n\pi\bigg| n\isin\mathbb{Z}\right\}.

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

\tan A = \frac {\textrm{opposite}} {\textrm{adjacent}} = \frac {a} {b}.

The set of zeroes of tangent is

\left\{n\pi\big| n\isin\mathbb{Z}\right\}.

The same set of the sine function since

\tan A = \frac {\sin A}{\cos A}.

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:

\csc A = \frac {\textrm{hypotenuse}} {\textrm{opposite}} = \frac {h} {a}.

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:

\sec A = \frac {\textrm{hypotenuse}} {\textrm{adjacent}} = \frac {h} {b}.

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:

\cot A = \frac {\textrm{adjacent}} {\textrm{opposite}} = \frac {b} {a}.

[edit] Definitions by power series

One can define the trigonometric functions also by using power series:

\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)!}
\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)!}
\tan x = \frac{\sin x}{\cos x}
\operatorname{cotan} x = \frac{\cos x}{\sin x}
\sec x = \frac{1}{\cos x}
\operatorname{cosec} x = \frac{1}{\sin x}.

[edit] References

[edit] See also

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[edit] External links

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