Editing Trigonometric functions (section)

== Periodicity and asymptotes ==
The sine and cosine functions are [[periodic function|periodic]], with period <math>2\pi</math>, which is the smallest positive period:
<math display="block">\sin(z+2\pi) = \sin(z),\quad \cos(z+2\pi) = \cos(z).</math>
Consequently, the cosecant and secant also have <math>2\pi</math> as their period. 

The functions sine and cosine also have semiperiods <math>\pi</math>, and
<math display="block">\sin(z+\pi)=-\sin(z),\quad \cos(z+\pi)=-\cos(z)</math>
and consequently
<math display="block">\tan(z+\pi) = \tan(z),\quad \cot(z+\pi) = \cot(z).</math>
Also, 
<math display="block">\sin(x+\pi/2)=\cos(x),\quad \cos(x+\pi/2) = -\sin(x)</math>
(see [[Angle#Combining_angle_pairs|Complementary angles]]).

The function <math>\sin(z)</math> has a unique zero (at <math>z=0</math>) in the strip <math>-\pi < \real(z) <\pi</math>. The function <math>\cos(z)</math> has the pair of zeros <math>z=\pm\pi/2</math> in the same strip. Because of the periodicity, the zeros of sine are 
<math display="block">\pi\mathbb Z = \left\{\dots,-2\pi,-\pi,0,\pi,2\pi,\dots\right\}\subset\mathbb C.</math>
There zeros of cosine are
<math display="block">\frac{\pi}{2} + \pi\mathbb Z = \left\{\dots,-\frac{3\pi}{2},-\frac{\pi}{2},\frac{\pi}{2},\frac{3\pi}{2},\dots\right\}\subset\mathbb C.</math>
All of the zeros are simple zeros, and both functions have derivative <math>\pm 1</math> at each of the zeros.

The tangent function <math>\tan(z)=\sin(z)/\cos(z)</math> has a simple zero at <math>z=0</math> and vertical asymptotes at <math>z=\pm\pi/2</math>, where it has a simple pole of residue <math>-1</math>. Again, owing to the periodicity, the zeros are all the integer multiples of <math>\pi</math> and the poles are odd multiples of <math>\pi/2</math>, all having the same residue. The poles correspond to vertical asymptotes
<math display="block">\lim_{x\to\pi^-}\tan(x) = +\infty,\quad \lim_{x\to\pi^+}\tan(x) = -\infty.</math>

The cotangent function <math>\cot(z)=\cos(z)/\sin(z)</math> has a simple pole of residue 1 at the integer multiples of <math>\pi</math> and simple zeros at odd multiples of <math>\pi/2</math>. The poles correspond to vertical asymptotes
<math display="block">\lim_{x\to 0^-}\cot(x) = -\infty,\quad \lim_{x\to 0^+}\cot(x) = +\infty.</math>