Understanding the Reciprocal Identity | csc(x) = sin(x) in Trigonometry

Reciprocal identity equal to csc(x)

The reciprocal identity equal to csc(x) is sin(x)

The reciprocal identity equal to csc(x) is sin(x).

In trigonometry, csc(x) is the abbreviation for the cosecant function. The cosecant function of an angle x is defined as the reciprocal of the sine function of that angle. Mathematically, we can represent this relationship as:

csc(x) = 1 / sin(x)

This means that the value of the cosecant of an angle x is equal to 1 divided by the value of the sine of that angle.

To understand this relationship, let’s consider an example.

Suppose we have an angle x in a right triangle. The sine of the angle x is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse.

sin(x) = Opposite side / Hypotenuse

Now, if we take the reciprocal of the sine function, we get

1 / sin(x) = Hypotenuse / Opposite side

This can be viewed as the ratio of the hypotenuse to the length of the side opposite to the angle x. This relationship is precisely the definition of the cosecant function, which is denoted as csc(x).

Therefore, the reciprocal identity equal to csc(x) is sin(x).

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