Question

NO LINKS!! Please help me with this problem Part 1e​

Answer 1

Step-by-step explanation:

`\displaystyle k = Acos(Bt - C) + D \\ \\ Vertical\:Shift \hookrightarrow D \\ Horisontal\:[Phase]\:Shift \hookrightarrow (C)/(B) \\ Wavelength\:[Period] \hookrightarrow (2)/(B)\pi \\ Amplitude \hookrightarrow |A| \\ \\ Vertical\:Shift \hookrightarrow 4 \\ Horisontal\:[Phase]\:Shift \hookrightarrow (C)/(B) \hookrightarrow 0 \\ Wavelength\:[Period] \hookrightarrow (2)/(B)\pi \hookrightarrow \boxed{24} \hookrightarrow (2)/((\pi)/(12))\pi \\ Amplitude \hookrightarrow 28`

First and foremost, you must move 28 to the left of the equivalence symbol to set the equation in its original standard formation. Doing so will give you the vertical shift of 4. This takes care of that.

Now, to explain why there is a double potential solution, trigonometric equations like this will ALWAYS have two solutions in the interval of their wavelengths, of which they are dubbed primary [or principal] and secondary solutions. Each solution will lie in unique quadrants. Now, to explain what
`\displaystyle n` symbolises, this is the easiest part because we cannot have a solution without knowing the wavelength of this equation, so we obviously need to find it, and by using the formula above, you arrive at twenty-four units. Therefore,
`\displaystyle n` represents the quantity of wavelengths in a given amount of time.

I am delighted to assist you at any time.

Answer 2

Each trigonometric function has two solutions in the interval of its period. As the given cosine function has a period of 24 units and therefore repeats every 24 units, we find the two solutions in the initial range (when n = 0) and add 24n to both of these. Therefore, the variable n represents the number of periods.

Step-by-step explanation:

Given equation:

`-28 \cos \left((\pi)/(12)t\right)+32=28`

Rearrange the given equation to a function:

`\implies -28 \cos \left((\pi)/(12)t\right)+32-28=28-28`

`\implies -28 \cos \left((\pi)/(12)t\right)+4=0`

`\implies f(x)=-28 \cos \left((\pi)/(12)t\right)+4`

Standard form of a cosine function

`\text{f}(x)=\text{A} \cos \left[\text{B}(x+\text{C})\right]+\text{D}`

where:

• A = amplitude (height from the midline to the peak).
• 2π/B = period (horizontal distance between consecutive peaks).
• C = phase shift (horizontal shift - positive is to the left).
• D = vertical shift.

Therefore, the period of the given function is:

`\sf B = (\pi)/(12) \implies Period=(2\pi)/((\pi)/(12))=24`

The cosine function is a periodic function. Therefore, the function repeats itself infinitely many times and thus there are an infinite number of solutions.

The period is the length of the shortest interval on the x-axis over which the graph repeats (the horizontal distance between consecutive peaks).

• The period of the given cosine function is 24.

Each trigonometric function has two solutions in the interval of its period.

• The first solution is the principal value.
• The other solution is called the secondary value and lies in a different quadrant.

As the given cosine function has a period of 24 units and therefore repeats every 24 units, we find the two solutions in the initial range and add 24n to both of these. Therefore:

• The variable n represents the number of periods.