Trigonometry

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# Secant

#### Lesson Contents

## Definition of Secant

In trigonometry, **the secant is the reciprocal of the cosine**. Secant is abbreviated as sec. The relation of secant and cosine is as follows:**sec(θ) = ^{1}⁄_{cos(θ)} and cos(θ) = ^{1}⁄_{sec(θ)}**

In a right triangle, the secant of an internal angle is the hypotenuse divided by the adjacent side, such that sec(θ) = ^{hypotenuse}⁄_{adjacent}.

### Secant's Inverse — sec^{-1} — Also Called Arcsecant

The inverse function of the secant is called arcsecant. In abbreviated form, this relation is given as:

**arcsec(θ) = sec(θ) ^{-1}**

The arcsecant follows the same relation as all other inverse trigonometric functions. It is the length that produces an angle where the sec of that angle is the length. Here is what the relation looks like when used:

sec(24°) = 1.095

arcsec(1.095) = 24°

### What Does the Graph Look Like?

The image below is what y = sec(x) looks like for a single period of the secant function.

Notice how the curves are asymptotic to x = π/2, x = 3π/2, and so forth? This is because at every half multiple of pi (excluding whole values of pi), the value of the function looks like:

y = sec(π/2) = ^{1}⁄_{cos(π/2)} = ^{1}⁄_{0}

Since dividing by zero is undefined, the function does not exist on those x values. The curve will go infinitely up or down in the vertical direction, but never touch the x values that it is asymptotic to.

Imagine going clockwise around the unit circle, starting at θ = 0 and ending at θ = π. The value of sec(θ) starts at 1, then approaches infinity as we approach θ = π/2. It becomes undefined at θ= π/2 and then becomes an infinitely large negative number just after θ = π/2. It then progresses to -1 at θ = π.