csc^(-1)x or arccsc(x)
The notation csc^(-1)x or arccsc(x) stands for the inverse cosecant function
The notation csc^(-1)x or arccsc(x) stands for the inverse cosecant function. It represents the angle (in radians or degrees) whose cosecant value is equal to x.
To understand this concept better, let’s start by reviewing the cosecant function (csc(x)). The cosecant of an angle is defined as the reciprocal of the sine of that angle. Mathematically, we can write it as:
csc(x) = 1 / sin(x)
The cosecant function takes an angle as input and gives the ratio of the length of the hypotenuse to the length of the side opposite to the angle in a right triangle. It is important to note that the cosecant function is only defined for angles where sine is nonzero.
Now, the inverse cosecant function (csc^(-1)x or arccsc(x)) works in the opposite way. Instead of taking an angle and giving a ratio, it takes a ratio (x) as input and gives the angle whose cosecant is equal to x.
Let’s illustrate this with an example. Suppose you are given the value of x as 2. To find the angle whose cosecant is 2, you would write it as:
csc^(-1)2
or
arccsc(2)
To solve this, you would find an angle (let’s call it θ) such that csc(θ) = 2. In other words:
csc(θ) = 2
1 / sin(θ) = 2
sin(θ) = 1/2
Since the sine of an angle is equal to 1/2, we know that this angle must be 30 degrees (or π/6 radians) because sine is positive in the first and second quadrants. Therefore, csc^(-1)2 or arccsc(2) is equal to 30 degrees (or π/6 radians).
It is worth mentioning that the cosecant function has a restricted domain. The cosecant of an angle is undefined (or infinite) when the sine of that angle is zero. Therefore, the inverse cosecant function is also restricted to specific intervals. Most commonly, the inverse cosecant function is defined for x values such that -∞ < x ≤ -1 or 1 ≤ x < +∞. I hope this explanation helped you understand the concept of csc^(-1)x or arccsc(x). If you have any further questions, feel free to ask!
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