The cosecant function (csc x) is a trigonometric function that is closely related to the sine (sin x), cosine (cos x), secant (sec x), and cotangent (cot x) functions. It is defined as the ratio of the length of the hypotenuse of a right triangle to the length of the opposite side. In other words, csc x = 1/sin x.
What is csc x?
The cosecant function, denoted as csc x, is the reciprocal of the sine function. It is defined as the ratio of the hypotenuse to the opposite side of a right triangle with an angle x.
Formula:
csc x = 1 / sin x
Domain and Range:
- Domain: All real numbers except for multiples of π (π, 2π, 3π, …)
- Range: All real numbers greater than or equal to 1 or less than or equal to -1
Properties:
- Odd function: csc (-x) = -csc x
- Periodic function with a period of 2π: csc (x + 2π) = csc x
- Asymptotes: Vertical asymptotes occur at x = nπ, where n is an integer.
Graph:
The graph of csc x has the following shape:
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Inverse Function:
The inverse function of csc x is the arcsine function, denoted as arcsin x.
Applications:
Csc x has applications in various fields, including:
- Trigonometry
- Physics
- Engineering
- Navigation
Examples:
- csc (π/3) = 2
- csc (-π/6) = -2
- csc (3π/4) = √2
Question 1: What is the definition of cosecant?
Answer: Cosecant, denoted as csc x, is a trigonometric function that represents the ratio of the length of the hypotenuse of a right triangle to the length of the side opposite to the angle x.
Question 2: What is the inverse function of csc x?
Answer: The inverse function of cosecant is arcsine, denoted as arcsin x, which calculates the angle whose cosecant is x.
Question 3: What is the range of csc x?
Answer: The range of cosecant is [-1, -∞] and [1, ∞], excluding zero. It assumes negative values for angles between -90° and -180° and between 90° and 180°, and positive values for angles between 0° and 90° and between -180° and -270°.
Well, there you have it! That was our quick dive into the definition and applications of csc x. I hope it helped shed some light on this fascinating trigonometric function. If you’ve got any more mathy questions, feel free to swing by again. I’ll be here, waiting to geek out on more trigonometry goodness with you. Thanks for reading, folks!