Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or trigonometric 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 most widely used trigonometric functions are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used in modern mathematics. Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.

See also - converter sin to cos.

The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extending these definitions to functions whose domain is the whole extended real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) is 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).

Examples:

1) The sine of angle a is the ratio of the opposite leg to the hypotenuse.

See online calculator - Sine Angle Calculator.

2) The cosine of angle a is the ratio of the adjacent leg to the hypotenuse.

See online calculator - Cosine Angle Calculator.

3) The tangent of an angle is a trigonometric function of an angle that is equal in a right-angled triangle to the ratio of a leg lying opposite a given acute angle to another leg.

See online calculator - Tangent Angle Calculator.

4) The cotangent of an angle is a trigonometric function of an angle, equal in a right triangle to the ratio of a leg, an adjacent angle, to another leg.

See online calculator - Cotangent Angle Calculator.

5) The secant of an angle is the trigonometric function of an angle, in a right triangle equal to the ratio of the hypotenuse to the leg adjacent to the given angle.

See online calculator - Secant Angle Calculator.

6) The cosecant of an angle is the trigonometric function of an angle, in a right triangle equal to the ratio of the hypotenuse to the leg opposite to the given angle.

See online calculator - Cosecant Angle Calculator.

Trigonometry on a circle is a pretty interesting abstraction in mathematics. If you understand the basic concept of the so-called "trigonometric circle", then all trigonometry will be subject to you.