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 } {\di...

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 ...

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  functi...