Determine the equation of a sine function that would have a range of {y| -4 ≤ y ≤ 1, y ε R} and a period of 45o.

Determine the cosine function that results in the same graph as the function above.
Deter

To find the equation of a sine function with the given range and period, we need to determine the amplitude, period, and vertical shift of the function.

1. Amplitude: The amplitude is half the range, so the amplitude is (1 - (-4))/2 = 5/2 = 2.5.

2. Period: The period is the length of one complete cycle of the function. Since the period is given as 45 degrees, we convert it to radians by multiplying by π/180. Therefore, the period is 45 * (π/180) = π/4.

3. Vertical Shift: The vertical shift is the average of the maximum and minimum values of the range. In this case, the average is (-4 + 1)/2 = -3/2 = -1.5.

The equation of the sine function can be written as: y = A * sin(Bx + C) + D, where:
- A is the amplitude
- B is the coefficient of x that determines the period
- C is a phase shift (which is not required in this case)
- D is the vertical shift

Plugging the values we determined into the equation, we have:

y = 2.5 * sin((π/4)x) - 1.5

To find the cosine function that results in the same graph, we use the identity: cos(x) = sin(x + π/2).

Therefore, the equation of the cosine function is:

y = 2.5 * cos((π/4)x) - 1.5

To determine the equation of a sine function with the given range and period, we need to analyze the characteristics of the sine function.

First, let's consider the amplitude of the function. The range given is -4 ≤ y ≤ 1. Since the amplitude is defined as half the difference between the maximum and minimum values, we can calculate:

Amplitude = (1 - (-4))/2 = 5/2 = 2.5.

So, the amplitude of the sine function is 2.5.

Next, let's focus on the period of the function. The period is the distance between two consecutive peaks or troughs. The given period is 45°. The period of a sine function can be calculated using the formula:

Period (in degrees) = 360° / Frequency,

where Frequency represents the number of complete cycles within 360°.

By rearranging the formula, we can determine the Frequency of the function:

Frequency = 360° / Period = 360° / 45° = 8.

Therefore, the Frequency of the sine function is 8.

Now, we have both the amplitude and frequency, so we can construct the equation of the sine function:

y = A * sin(Bx + C),

where A represents the amplitude, B is the frequency, and C is the phase shift.

Plugging in the values, we get:

y = 2.5 * sin(8x + C).

Now, we need to determine the phase shift, C, to match the given range.

The minimum value on the y-axis is -4. Since the standard sine function starts at 0, we need to find the phase shift that brings the minimum value down to -4. We can do this by subtracting the phase shift from the angle where the standard sine function achieves its minimum.

The standard sine function achieves its minimum at x = 0. Therefore, we have:

y = 2.5 * sin(8(0) + C) = -4.

Simplifying, we get:

sin(C) = -4/2.5 = -1.6.

However, the range of the sine function is between -1 and 1, so there is no real angle that satisfies this equation. Hence, the given range of {-4 ≤ y ≤ 1} cannot be represented by a sine function with a period of 45°.

To determine the cosine function that results in the same graph as the function described, we need to use the cosine function:

y = A * cos(Bx + C).

Since cosine is just a phase-shifted version of sine, the cosine function with the same graph as the desired function can be obtained by just changing the phase shift appropriately.

For sine functions, the phase shift is π/2 radians (or 90°) to the right:

C = π/2.

Therefore, the equation of the cosine function that results in the same graph is:

y = 2.5 * cos(8x + π/2).

The midpoint of the range -4 <= y <=1 is y = -1.5 with an amplitude of 2.5

The equation is now of the form:

y = 2.5*sin(a*x) - 1.5

When a = 1, the period is 360 degrees. So when a = 45/360; the period will be 45 degrees, or when a = 1/8

y = 2.5*sin(x/8) - 1.5

for the second part, note that sin(a*x) = cos(a*(x-90))

y = 2.5*sin((x-90)/8) -1.5