To determine a sine function that has its first minimum at $(\pi/3, 3)$ and its first maximum at $(4\pi/3, 9)$, let's start by analyzing the general form of a sine function:

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

where:

• $A$ is the amplitude,
• $B$ affects the period of the function,
• $C$ is the phase shift,
• $D$ is the vertical shift.

Given the points:

• Minimum at $(\pi/3, 3)$
• Maximum at $(4\pi/3, 9)$

Step-by-Step Solution:

1. Determine the Vertical Shift ($D$):

The average of the maximum and minimum values gives the vertical shift: $D = \frac{9 + 3}{2} = 6$

2. Determine the Amplitude ($A$):

The amplitude is half the distance between the maximum and minimum values: $A = \frac{9 - 3}{2} = 3$

3. Determine the Period and $B$:

The sine function completes one full cycle between the minimum and the next minimum (or maximum to maximum). From $\pi/3$ to $4\pi/3$ (distance of $4\pi/3 - \pi/3 = \pi$) is half the period. Hence, the full period $T$ is $2\pi$: $T = 2 \times (\frac{4\pi/3 - \pi/3}{1/2}) = 2\pi$ Thus, $B = \frac{2\pi}{T} = 1$

4. Determine the Phase Shift ($C$):

Since the function has its first minimum at $(\pi/3, 3)$, we need to shift the basic sine function accordingly. A sine function has its minimum at $3\pi/2$ (without any shifts). Thus, we solve for $C$ in: $B \left(\frac{\pi}{3}\right) + C = \frac{3\pi}{2}$ $\frac{\pi}{3} + C = \frac{3\pi}{2}$ $C = \frac{3\pi}{2} - \frac{\pi}{3}$ $C = \frac{9\pi - 2\pi}{6}$ $C = \frac{7\pi}{6}$

Therefore, the sine function is: $y = 3 \sin(x - \frac{7\pi}{6}) + 6$

Equivalent Cosine Function:

A cosine function can be written in the form: $y = A \cos(Bx + C) + D$

The cosine function can be converted from the sine function using the phase shift: $\cos(x) = \sin(x + \frac{\pi}{2})$

So the equivalent cosine function is: $y = 3 \cos(x - \frac{7\pi}{6} - \frac{\pi}{2}) + 6$ $y = 3 \cos(x - \frac{7\pi + 3\pi}{6}) + 6$ $y = 3 \cos(x - \frac{10\pi}{6}) + 6$ $y = 3 \cos(x - \frac{5\pi}{3}) + 6$

Thus, the cosine function is: $y = 3 \cos(x - \frac{5\pi}{3}) + 6$

Summary:

a. The sine function is: $y = 3 \sin(x - \frac{7\pi}{6}) + 6$

b. The equivalent cosine function is: $y = 3 \cos(x - \frac{5\pi}{3}) + 6$

Would you like any further details or have any questions?

Here are some related questions for you to consider:

1. How do you determine the phase shift in a trigonometric function?
2. What is the significance of the amplitude in a sine or cosine function?
3. How do you find the period of a sine or cosine function?
4. Can you explain the difference between a sine and a cosine function in terms of their graphs?
5. How does the vertical shift affect the graph of a sine or cosine function?
6. What happens to the graph of a sine function if you change the amplitude?
7. How do you convert a sine function to a cosine function and vice versa?
8. What is the impact of the phase shift on the starting point of a sine or cosine wave?

Tip: Remember that the phase shift in a sine or cosine function shifts the graph horizontally, changing where the function starts its cycle.