To write the equation of a sine function with the given characteristics:

• Midline: $y = 4$
• Amplitude: $3$
• Period: $\frac{5\pi}{3}$

The general form of a sine function is:

$y = A \sin(Bx + C) + D$

Where:

• $A$ is the amplitude.
• $B$ is related to the period by the formula $\text{Period} = \frac{2\pi}{B}$.
• $D$ is the vertical shift (midline).

Given the information:

• $A = 3$
• The period is $\frac{5\pi}{3}$, so $B$ can be found using $\frac{2\pi}{B} = \frac{5\pi}{3}$. Solving this gives $B = \frac{6}{5}$.
• $D = 4$

Since no horizontal shift is mentioned, we assume $C = 0$.

So the equation of the sine function is:

$y = 3 \sin\left(\frac{6}{5}x\right) + 4$

Would you like to go into more details or have any questions?

Here are 5 related questions for you to explore further:

1. How would the function change if the period were doubled?
2. What is the phase shift if $C$ were $\frac{\pi}{4}$?
3. How would you write the equation for a cosine function with the same properties?
4. What are the maximum and minimum values of this function?
5. How do you determine the phase shift from a given function?

Tip: When writing trigonometric functions, always check if the equation needs any horizontal or vertical shifts!