Secant Function: Definition, Properties, And Range

reciprocal cosx

1/secx

The reciprocal of cosine is called secant and is denoted by sec(x). It is defined as the ratio of the hypotenuse to the adjacent side of a right-angled triangle that contains the angle x.

We can express sec(x) in terms of cosine as follows:

sec(x) = 1/cos(x)

That is, sec(x) is the multiplicative inverse or reciprocal of cosine, which means that sec(x) always has the opposite sign to cosine, except at the points where cosine equals zero.

The domain of sec(x) is the set of all real numbers except the points where cosine equals zero, i.e., x = (2n + 1)π/2, where n is any integer. At these points, sec(x) is undefined since the denominator becomes zero.

Some important properties of sec(x) are:

1. Even function: sec(x) = sec(-x), which means that sec(x) is symmetrical about the y-axis.

2. Periodicity: sec(x) has a period of 2π, which means that the graph of sec(x) repeats itself every 2π units.

3. Divergence: sec(x) is not defined at the points where cosine equals zero, and it approaches positive or negative infinity as x approaches these points from either side.

4. Range: The range of sec(x) is (-∞, -1] ∪ [1, ∞), which means that sec(x) can take any value greater than or equal to 1 or less than or equal to -1.

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