Let's find the equation of the cosecant function that satisfies the given conditions.
### Step 1: Write the General Form
The general form of the cosecant function is:
[tex]\[ y = A \cdot \csc(Bx - C) + D \][/tex]
where:
- [tex]\( A \)[/tex] is the amplitude,
- [tex]\( B \)[/tex] affects the period,
- [tex]\( C \)[/tex] is the phase shift,
- [tex]\( D \)[/tex] is the vertical shift.
### Step 2: Determine the Amplitude ([tex]\( A \)[/tex])
The range of the cosecant function is [tex]\( (-\infty, -7] \cup [7, \infty) \)[/tex]. This implies that the amplitude is 7, since the distance from the center (midpoint) to the maximum and minimum values is 7. Thus:
[tex]\[ A = 7 \][/tex]
### Step 3: Determine the Period and [tex]\( B \)[/tex]
The period of the given function is [tex]\( 5\pi \)[/tex]. The period ([tex]\( T \)[/tex]) and [tex]\( B \)[/tex] are related by the formula:
[tex]\[ T = \frac{2\pi}{B} \][/tex]
Given [tex]\( T = 5\pi \)[/tex], we can solve for [tex]\( B \)[/tex] as follows:
[tex]\[ 5\pi = \frac{2\pi}{B} \][/tex]
[tex]\[ B = \frac{2\pi}{5\pi} \][/tex]
[tex]\[ B = \frac{2}{5} \][/tex]
### Step 4: Determine the Phase Shift ([tex]\( C \)[/tex])
We are given that there is no information on the phase shift of the function, so we assume:
[tex]\[ C = 0 \][/tex]
### Step 5: Determine the Vertical Shift ([tex]\( D \)[/tex])
Since there is no vertical shift indicated, the vertical shift [tex]\( D \)[/tex] is 0:
[tex]\[ D = 0 \][/tex]
### Step 6: Write the Equation
Substituting [tex]\( A = 7 \)[/tex], [tex]\( B = \frac{2}{5} \)[/tex], [tex]\( C = 0 \)[/tex], and [tex]\( D = 0 \)[/tex] into the general form, we get:
[tex]\[ y = 7 \cdot \csc\left(\frac{2}{5}x\right) \][/tex]
Thus, the equation for the cosecant function is:
[tex]\[ y = 7 \cdot \csc\left(\frac{2}{5}x\right) \][/tex]
This is the simplified equation of the given cosecant function.
