Understanding the Cosecant Function: Periodicity, Range, Symmetry, Asymptotes, and Zeros.

f(x) = csc x

f'(x) = -csc x • cot x

The function f(x) = csc x represents the cosecant function, which is the reciprocal of the sine function. It is defined as:

csc x = 1/sin x

The function is defined for all values of x except for those where sin x = 0, which occur at multiples of pi (i.e. x = n*pi, where n is an integer).

Some key features of the cosecant function:

1. Periodicity: The function has a period of 2 pi. This means that the function repeats itself after every 2 pi units of x.

2. Range: The range of the function is the set of all real numbers except for values between -1 and 1, since sin x can only take values between -1 and 1. Therefore, the range of csc x is (-infinity, -1] U [1, infinity).

3. Symmetry: The function is symmetric about the vertical line x = pi/2. This means that csc (pi/2 + x) = csc (pi/2 – x), or graphically, the function is reflected around the line x = pi/2.

4. Asymptotes: The function has vertical asymptotes where sin x = 0, i.e. at x = n*pi where n is an integer. The function approaches infinity as x approaches these asymptotes from either direction.

5. Zeros: The function has zeros where sin x = 1, i.e. at x = (2n + 1) * pi/2 where n is an integer.

To graph the function, we can start by plotting the asymptotes at x = n*pi. We can also plot the zeros at x = (2n + 1) * pi/2. Then, we can plot points using values from the function’s range. For example, we can calculate csc (0) = 1/sin(0) = infinity, csc (pi/2) = 1/sin (pi/2) = 1, csc (pi) = 1/sin (pi) = -infinity, etc. We can then connect these points to create a smooth curve.

Overall, the cosecant function is a periodic function with vertical asymptotes and zeros. It is defined for all values of x except where sin x = 0, and its range is (-infinity, -1] U [1, infinity).

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