In this problem, you will describe in detail how we arrive at the accepted definition of the function y= Arccsc(x)
a) Sketch the cosecant function over the interval (-2pi, 2pi)
i can do that
b) This function is not one to one - please explain in detail the meaning of this
???
c) how would you restrict the domain of the cosecant function so that it becomes a one to one function?
???
d) By restricting the function as described in part c) what is the domain and range of this resulting function. Use interval notation.
e) From part d) what is the domain and range of the inverse function
y = arccsc(x).
After you graph it you will find that more than one angle has the same csc. For example csc 30 deg = 2 and csc 150 deg is also 2
if you restrict the domain to one quadrant you are all set.
a) To sketch the cosecant function over the interval (-2π, 2π), we need to understand what the cosecant function represents. The cosecant function, denoted as csc(x), is the reciprocal of the sine function. In other words, csc(x) = 1/sin(x).
To sketch the graph, we start by plotting some key points. The important points to consider are where the sine function equals zero, as the cosecant function will be undefined at those points. The sine function equals zero at x = 0, π, and 2π.
Next, we need to determine the behavior of the function between those points. Since the sine function has a period of 2π, we can repeat the pattern of the cosecant function beyond those points. We observe that cosecant is positive between 0 and π and negative between π and 2π.
Finally, we draw a smooth curve passing through these points and respecting the behavior of the function over the entire interval (-2π, 2π).
b) The cosecant function is not one-to-one. This means that for certain values of x, there can be more than one value of y. In other words, there are multiple points on the graph of the cosecant function that have the same y-value. This happens because the cosecant value can repeat periodically, leading to multiple solutions for the same value of y.
c) To restrict the domain of the cosecant function so that it becomes a one-to-one function, we can choose a smaller interval where the function doesn't repeat. One such interval is [-π/2, π/2].
In this interval, the cosecant function is always positive since the sine function is positive for all angles. This ensures that there are no repeats or multiple values for the same y. Restricting the domain in this way allows us to make the cosecant function one-to-one over this specific interval.
d) By restricting the function as described in part c, the domain of the resulting function (cosecant restricted to [-π/2, π/2]) is [-π/2, π/2]. The range, or the set of all possible output values, of this function is (-∞, -1] U [1, ∞). In interval notation, the domain is [-π/2, π/2] and the range is (-∞, -1] U [1, ∞).
e) Since the function y = arccsc(x) is the inverse of the restricted cosecant function, the domain and range will be reversed. Therefore, the domain of the inverse function is (-∞, -1] U [1, ∞) and the range is [-π/2, π/2]. So, in interval notation, the domain of the inverse function is (-∞, -1] U [1, ∞) and the range is [-π/2, π/2].