Write an equation for a cosecant function with a period of 3𝜋 and a range of (−�‡,−2]�¾[2,�‡)
so you want y = a csc (2/3)x
where a is whatever your (−�‡,−2]�¾[2,�‡) means
well, y = csc (2x/3) looks like this:
http://www.wolframalpha.com/input/?i=plot+y+%3D+csc+(2x%2F3)
the vertical lines are asymptotes at
x = 0, ±3π/2, ± 3π, ...
the range is y ≥1 OR y ≤ -1
To write the equation for a cosecant function with a period of 3𝜋 and a range of (−�‡,−2]�¾[2,�‡), we need to consider the properties of the cosecant function.
The cosecant function (csc𝜃) is defined as the reciprocal of the sine function (sin𝜃). It represents the ratio of the hypotenuse to the length of the side opposite an angle in a right triangle.
The general equation for a cosecant function is given by:
csc𝜃 = A * sin(B𝜃 + C) + D
where A, B, C, and D are constants that affect the amplitude, period, phase shift, and vertical shift of the function, respectively.
Given that the period is 3𝜋, we can determine the value of B as follows:
Period = 2𝜋 / |B|
Substituting the value of the period (3𝜋) into this equation, we get:
3𝜋 = 2𝜋 / |B|
Simplifying further:
|B| = 2/3
Since 𝜋 is a positive value, we can conclude that B = 2/3.
Next, let's consider the range of (−�‡,−2]�¾[2,�‡). The range of the cosecant function is all real numbers except for where the sine function equals zero. In other words, the range of csc𝜃 is (-∞, -1] U [1, +∞).
However, we need to adjust this range to match the given range of (-�‡,−2]�¾[2,�‡). To do this, we can apply vertical shifts to the cosecant function. Since the range is shifted down by 1 unit, we can set D = -1.
Putting it all together, the equation for the cosecant function with a period of 3𝜋 and a range of (−�‡,−2]�¾[2,�‡) is:
csc𝜃 = A * sin((2/3)𝜃) - 1
Please note that the value of A has not been specified. The value of A can be determined based on the desired amplitude of the function.