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 as such are also widely used for studying periodic phenomena through Fourier analysis.

The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and 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 these definitions to functions whose domain is the whole projectively extended real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used. 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 from which some isolated points are removed.

Right-angled triangle definitions

In this right triangle: $sinA={\frac {a}{c}}$; $cosA={\frac {b}{c}}$; $tanA={\frac {a}{b}}$.

In this section, an upper-case letter denotes a vertex of a triangle and the measure of the corresponding angle; The lower-case form of the same letter denotes the opposite side of the triangle and its length. In the definitions that follow, θ corresponds to A in the diagram.

If the angle θ is given, then all sides of the right-angled triangle are well-defined up to a scaling factor. 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. More precisely, the six trigonometric functions are:

sine $\sin \theta ={\frac {a}{c}}={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}$	cosecant $\csc \theta ={\frac {c}{a}}={\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}$
cosine $\cos \theta ={\frac {b}{c}}={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}$	secant $\sec \theta ={\frac {c}{b}}={\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}$
tangent $\tan \theta ={\frac {a}{b}}={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}$	cotangent $\cot \theta ={\frac {b}{a}}={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}$

In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or ${\frac {\pi }{2}}$ radians.

Summary of relationships between trigonometric functions

Function	Abbreviation	Description	Relationship
			using radians	using degrees
sine	sin	${\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}$	$\sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }}$	$\sin x=\cos \left(90^{\circ }-x\right)={\frac {1}{\csc x}}$
cosine	cos	${\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}$	$\cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sec \theta }}\,$	$\cos x=\sin \left(90^{\circ }-x\right)={\frac {1}{\sec x}}\,$
tangent	tan (or tg)	${\frac {\mathrm {opposite} }{\mathrm {adjacent} }}$	$\tan \theta ={\frac {\sin \theta }{\cos \theta }}=\cot \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cot \theta }}$	$\tan x={\frac {\sin x}{\cos x}}=\cot \left(90^{\circ }-x\right)={\frac {1}{\cot x}}$
cotangent	cot (or cotan or cotg or ctg or ctn)	${\frac {\mathrm {adjacent} }{\mathrm {opposite} }}$	$\cot \theta ={\frac {\cos \theta }{\sin \theta }}=\tan \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\tan \theta }}$	$\cot x={\frac {\cos x}{\sin x}}=\tan \left(90^{\circ }-x\right)={\frac {1}{\tan x}}$
secant	sec	${\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}$	$\sec \theta =\csc \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cos \theta }}$	$\sec x=\csc \left(90^{\circ }-x\right)={\frac {1}{\cos x}}$
cosecant	csc (or cosec)	${\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}$	$\csc \theta =\sec \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sin \theta }}$	$\csc x=\sec \left(90^{\circ }-x\right)={\frac {1}{\sin x}}$

Trigonometric functions on the unit circle

In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane.

The unit circle, with some points labeled with their cosine and sine (in this order), and the corresponding angles in radians and degrees.

The trigonometric functions cosine and sine of angle θ may be defined on the unit circle as follows: If (x, y) is a point on the unit circle, and if the ray from the origin (0, 0) to (x, y) makes an angle θ from the positive x-axis, (where counterclockwise turning is positive), then

$\cos \theta =x\quad {\text{and}}\quad \sin \theta =y.$

The equation $x^{2}+y^{2}=1$ gives the relation

$\cos ^{2}\theta +\sin ^{2}\theta =1.$

The unit circle also demonstrates that sine and cosine are periodic functions, with the identities

$\cos \theta =\cos(2\pi k+\theta )$

$\sin \theta =\sin(2\pi k+\theta )$

for any integer k.

Resources

• Trig Functions, Paul's Online Notes
• Trig Cheat Sheet, Paul's Online Notes

References

1. Klein, Christian Felix (1924) [1902]. Elementarmathematik vom höheren Standpunkt aus: Arithmetik, Algebra, Analysis (in German). 1 (3rd ed.). Berlin: J. Springer.
2. Klein, Christian Felix (2004) [1932]. Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis. Translated by Hedrick, E. R.; Noble, C. A. (Translation of 3rd German ed.). Dover Publications, Inc. / The Macmillan Company. ISBN 978-0-48643480-3. Archived from the original on 2018-02-15. Retrieved 2017-08-13.
3. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-29.
4. Protter, Murray H.; Morrey, Charles B., Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, LCCN 76087042