ratios of two side lengths no relation cotangent
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 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.Periods
All trigonometric functions are periodic functions of period 2π. This is the smallest period, except for the tangent and the cotangent, which have π as smallest period. This means that, for every integer k, one hassin
                ⁡
                (
                x
                +
              
              
                2
                k
                π
                )
              
              
                =
                sin
                ⁡
                x
              
            
            
              
                cos
                ⁡
                (
                x
                +
              
              
                2
                k
                π
                )
              
              
                =
                cos
                ⁡
                x
              
            
            
              
                tan
                ⁡
                (
                x
                +
              
              
                k
                π
                )
              
              
                =
                tan
                ⁡
                x
              
            
            
              
                cot
                ⁡
                (
                x
                +
              
              
                k
                π
                )
              
              
                =
                cot
                ⁡
                x
              
            
            
              
                csc
                ⁡
                (
                x
                +
              
              
                2
                k
                π
                )
              
              
                =
                csc
                ⁡
                x
              
            
            
              
                sec
                ⁡
                (
                x
                +
              
              
                2
                k
                π
                )
              
              
                =
                sec
                ⁡
                x
                .
              
            
          
        
      
    
    {\displaystyle {\begin{array}{lrl}\sin(x+&2k\pi )&=\sin x\\\cos(x+&2k\pi )&=\cos x\\\tan(x+&k\pi )&=\tan x\\\cot(x+&k\pi )&=\cot x\\\csc(x+&2k\pi )&=\csc x\\\sec(x+&2k\pi )&=\sec x.\end{array}}}
no relation