Na Galipedia, a Wikipedia en galego.
A cotanxente, co símbolo habitual cot ou cotan, é unha función trigonométrica.
Gráfica da función cotanxente.
Xeométricamente, nun triángulo rectángulo ABC con hipotenusa AB :
![{\displaystyle \cot {\hat {A}}=\mathrm {\frac {AC}{BC}} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BatGOotrDyjs4aNm1zgs5zNeNzNK2ntmOoqnDaNe1nje3aNmNatCP)
En trigonometría :
![{\displaystyle \cot \theta ={\cos \theta \over \sin \theta }={1 \over \tan \theta }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CzDaQnjKQatrAotKPyjG3ztaPygaPo2vAnDa1zNe4ztw0njw1z2aN)
A función cotanxente verifica a igualdade :
![{\displaystyle 1+\cot ^{2}x=\csc ^{2}x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CotoPnjaPajGOzjJFzqe1zNvFytm1yjBDoAvBoDdBzgo2zqdAoAiQ)
A derivada da cotanxente é :
![{\displaystyle \cot 'x=-\csc ^{2}x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AzNi0aqaPotJBaDvFzqiNzNhFa2s0o2zBoDdAaAaOotrFytm0zgs4)
A antiderivada da cotanxente é :
![{\displaystyle \int \cot x\,\mathrm {d} x=\ln(\sin x)+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OaDe0zDmPzAo0aAnEnjK5a2sNa2a5yjrAoDiOajvEajzFati5oqvB)
Temos a expansión da serie de Laurent, onde
designa o k-ésimo número de Bernoulli
![{\displaystyle \cot x=\sum _{n=0}^{+\infty }{\frac {(-1)^{n}2^{2n}B_{2n}}{(2n)!}}x^{2n-1}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QoNK0zAo1a2vFatKOnta2yqsOzNG3zAeQoDKQaNo3oNBFztrDzAvD)
máis tamén
![{\displaystyle \pi \cot(\pi x)={\frac {1}{x}}+\sum _{n=1}^{+\infty }{\frac {2x}{x^{2}-n^{2}}}=\sum _{n=-\infty }^{+\infty }{\frac {1}{x+n}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85otlEaNs4zAaOotCNntw2yqwPygvDnjvBatm4otnByqdDnDw0ztK2)
do que deducimos
![{\displaystyle \cot(x)={\frac {1}{x}}+\sum _{n=1}^{+\infty }{\frac {2x}{x^{2}-n^{2}\pi ^{2}}}=\sum _{n=-\infty }^{+\infty }{\frac {1}{x+n\pi }}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82nje3o2a4agePoDCPatJCaNnFytFDyjzDzAo0ygsQzDC5ateQnjhC)
Weisstein, Eric W. "Cotangent". MathWorld.