Understanding the Cosecant Function (csc(x)) and Its Properties in Mathematics

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The function csc(x) stands for cosecant of x, and it is the reciprocal of the sine function

The function csc(x) stands for cosecant of x, and it is the reciprocal of the sine function. In other words, csc(x) = 1/sin(x).

To understand this function better, we need to know some properties of sine. The sine function measures the ratio of the length of the side opposite an angle in a right triangle to the length of the hypotenuse. It ranges from -1 to 1, inclusive, and has periodic behavior with a period of 2π.

Now, let’s discuss the properties of the cosecant function, csc(x):

1. Domain: The domain of csc(x) is the set of all real numbers except for those values for which sin(x) is zero, because division by zero is undefined. Therefore, the domain can be expressed as x ≠ kπ, where k is any integer.

2. Range: The range of csc(x) is the set of all real numbers except for values between -1 and 1, inclusive. In other words, the range of csc(x) is (-∞, -1) ∪ (1, ∞).

3. Graph: The graph of csc(x) has a vertical asymptote at x = kπ, where k is any integer, because sin(x) is zero at these points. The graph also has positive and negative branches that approach these asymptotes as x approaches them.

4. Periodicity: The cosecant function, csc(x), is periodic with a period of 2π, just like sine. This means that csc(x) = csc(x + 2π) = csc(x + 4π), and so on.

To find the value of csc(x) for a specific angle x, you can calculate sin(x), then take the reciprocal of that value.

For example, if we need to find csc(π/6), we know that sin(π/6) = 1/2. So, csc(π/6) = 1/(1/2) = 2.

If we wanted to find csc(3π/4), we can determine that sin(3π/4) = √2/2. Therefore, csc(3π/4) = 1/(√2/2) = 2/√2 = √2.

I hope this explanation helps you understand the concept of csc(x) and how to calculate its value for different angles. If you have any further questions, please feel free to ask!

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