We have 6 trigonometric functions: sin, cos, tan, sec, csc and cot.
The first two, sine and cosine, are the fundamental ones. They have the same shape but are out of phase by π/2 radians.
Then, the tangent function is equal to the quotient sin/cos. Then, it has discontinuities everytime the cosine function is equal to 0.
The cotangent function is the inverse of the tangent function. Then, it is undefined everytime the tangent function is 0. This correspond to the values where the sine function is 0.
The secant function is the inverse of the cosine function. It has discontinuities in the same values as the tangent.
The cosecant function is the inverse of the sine function, and it is out of phase from the secant by π/2 radians.
All of the six functions have a period of 2π when the argument is x.