Mathematics

Trigonometric functions of real or complex variables Main article:  Trigonometric function Using the  unit circle , one can extend the definitions of trigonometric ratios to all positive and negative arguments [36]  (see  trigonometric function ). Graphs of trigonometric functions The following table summarizes the properties of the graphs of the six main trigonometric functions: [37] [38] Function Period Domain Range Graph sine {\displaystyle 2\pi } {\displaystyle (-\infty ,\infty )} {\displaystyle [-1,1]} cosine {\displaystyle 2\pi } {\displaystyle (-\infty ,\infty )} {\displaystyle [-1,1]} tangent {\displaystyle \pi } {\displaystyle x\neq \pi /2+n\pi } {\displaystyle (-\infty ,\infty )} secant {\displaystyle 2\pi } {\displaystyle x\neq \pi /2+n\pi } {\displaystyle (-\infty ,-1]\cup [1,\infty )} cosecant {\displaystyle 2\pi } {\displaystyle x\neq n\pi } {\displaystyle (-\infty ,-1]\cup [1,\infty )} cotangent {\displaystyle \pi } {\displaystyle x\neq n\

The  reciprocals  of these functions are named the  cosecant  (csc),  secant  (sec), and  cotangent  (cot), respectively: {\displaystyle \csc A={\frac {1}{\sin A}}={\frac {\textrm {hypotenuse}}{\textrm {opposite}}}={\frac {c}{a}},} {\displaystyle \sec A={\frac {1}{\cos A}}={\frac {\textrm {hypotenuse}}{\textrm {adjacent}}}={\frac {c}{b}},} {\displaystyle \cot A={\frac {1}{\tan A}}={\frac {\textrm {adjacent}}{\textrm {opposite}}}={\frac {\cos A}{\sin A}}={\frac {b}{a}}.} The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-". [30] With these functions, one can answer virtually all questions about arbitrary triangles by using the  law of sines  and the  law of cosines . [31]  These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known

Trigonometric ratios Main article:  Trigonometric function In this right triangle:  sin  A  =  a / c ;   cos  A  =  b / c ;   tan  A  =  a / b . Trigonometric ratios are the ratios between edges of a right triangle. These ratios are given by the following  trigonometric functions  of the known angle  A , where  a ,  b  and  c  refer to the lengths of the sides in the accompanying figure: Sine  function (sin), defined as the ratio of the side opposite the angle to the  hypotenuse . {\displaystyle \sin A={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}={\frac {a}{c}}.} Cosine  function (cos), defined as the ratio of the  adjacent  leg (the side of the triangle joining the angle to the right angle) to the hypotenuse. {\displaystyle \cos A={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}={\frac {b}{c}}.} Tangent  function (tan), defined as the ratio of the opposite leg to the adjacent leg. {\displaystyle \tan A={\frac {\textrm {opposite}}{\textrm {