SEC X, or secant of x, is a fundamental trigonometric function. It's defined as the reciprocal of the cosine function.
Defining Secant
- Formula:
sec x = 1 / cos x
This means that the secant of an angle x is equal to 1 divided by the cosine of that angle. Since the cosine function represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle, the secant represents the ratio of the hypotenuse to the adjacent side.
Understanding the Relationship with Cosine
The secant function's value is directly dependent on the cosine function's value.
- When cos x is close to 0: sec x approaches positive or negative infinity. This results in vertical asymptotes in the graph of the secant function.
- When cos x = 1: sec x = 1.
- When cos x = -1: sec x = -1.
Domain and Range
As noted in the Dartmouth notes (https://math.dartmouth.edu/opencalc2/cole/lecture17.pdf), the secant and tangent functions share the same domains. The range of sec x is more complex due to the bounds of the cosine function (-1 ≤ cos x ≤ 1). Therefore, sec x is never between -1 and 1.
Practical Applications
The secant function, like other trigonometric functions, has various applications in:
- Physics: Calculating angles and distances in projectile motion or wave mechanics.
- Engineering: Designing structures, analyzing forces, and modeling periodic phenomena.
- Navigation: Determining positions and distances using triangulation.
