Sinusoidal Functions - Sine Regression Function

[hivar]

Hi all. I'm currently having trouble with sinusoidal functions when I'm using the TI-84 Plus calculator.

I was taught that the equation of a sinusoidal function is y = a sin b (x - c) + d, however, when I use SinReg on the TI-84 Plus calculator, the formula is listed as y = a*sin(bx+c)+d.

I am confused about "c" when it is a negative. If the calculator spits out a negative number with "c", does that mean the equation for that problem is turned into y = a sin b (x + c) + d?

I have two example problems here with conflicting sinusoidal regression equations.

Ex 1. The following table shows the number of hours of daylight for an area in {location} on certain days of the year 2011.

Date​	Day Number​	Hours of Daylight​
January 1​	1​	8.00​
February 12​	43​	9.87​
March 15​	74​	11.83​
April 27​	117​	14.59​
May 8​	128​	15.20​
June 28​	179​	16.53​
July 18​	199​	16.00​
August 10​	222​	14.87​
September 8​	251​	13.10​
October 17​	290​	10.62​
November 6​	310​	9.42​
December 22​	356​	7.90​

Q: What is a sine regression function that gives the number of hours of daylight, H(d), with respect to the day number, d, of the year 2011? Round the values of a, c and to the nearest hundredth and round the value for b to the nearest thousandth.

The sine regression function given in the book is H(d) = 4.23sin(0.017d - 1.30) + 12.12.

Ex 2:

A wind turbine has four blades mounted on a concrete tower. The following table shows the height of the tip of one of the blades of the turbine as it turns through two rotations.

Time (s)​	Height (m)​
1​	10.2​
2.5​	9.1​
4​	4.8​
5.5​	5.9​
7.5​	10.7​
9​	7.5​
10.5​	4.4​

Q: What is a sine function that represents the height above the ground, h(t), of the tip of the blade with respect to time, t, in seconds? Round the values of a, b, c and d to the nearest hundredth.

The sine regression function given in the book is h(t) = 3.15sin(1.05d + 0.02) + 7.51.

Both example problems gave negative "c" answers, yet one problem has c being added to b and the other is subtracted from b. What is the correct answer here and moving forward, when asked to find sine regression functions on the calculator, if the value of "c" is negative, what would the sine regression function look like?

Thanks.

 

[stapel]

I was taught that the equation of a sinusoidal function is y = a sin b (x - c) + d, however, when I use SinReg on the TI-84 Plus calculator, the formula is listed as y = a*sin(bx+c)+d.

They are equivalent forms:

[imath]\qquad a \sin(b(x-c)) + d[/imath]

[imath]\qquad a \sin(bx - bc) + d[/imath]

[imath]\qquad a \sin(bx + C) + d[/imath]

...where [imath]-bc = +C[/imath]

I have two example problems here with conflicting sinusoidal regression equations.

Ex 1. The following table shows the number of hours of daylight for an area in {location} on certain days of the year 2011.

Date​	Day Number​	Hours of Daylight​
January 1​	1​	8.00​
February 12​	43​	9.87​
March 15​	74​	11.83​
April 27​	117​	14.59​
May 8​	128​	15.20​
June 28​	179​	16.53​
July 18​	199​	16.00​
August 10​	222​	14.87​
September 8​	251​	13.10​
October 17​	290​	10.62​
November 6​	310​	9.42​
December 22​	356​	7.90​

Q: What is a sine regression function that gives the number of hours of daylight, H(d), with respect to the day number, d, of the year 2011? Round the values of a, c and to the nearest hundredth and round the value for b to the nearest thousandth.

The sine regression function given in the book is H(d) = 4.23sin(0.017d - 1.30) + 12.12.

The generic sine curve starts at the origin (in particular, at [imath]y=0[/imath]) and moved upward and then downward over its first half-period. This is similar to the values in your "Hours" column. There appears to be a *slight* offset to the right, which led to the [imath]-1.30[/imath].

By the way, what equation did *you* get? By what steps?

Ex 2: A wind turbine has four blades mounted on a concrete tower. The following table shows the height of the tip of one of the blades of the turbine as it turns through two rotations.

Time (s)​	Height (m)​
1​	10.2​
2.5​	9.1​
4​	4.8​
5.5​	5.9​
7.5​	10.7​
9​	7.5​
10.5​	4.4​

Q: What is a sine function that represents the height above the ground, h(t), of the tip of the blade with respect to time, t, in seconds? Round the values of a, b, c and d to the nearest hundredth.

The sine regression function given in the book is h(t) = 3.15sin(1.05d + 0.02) + 7.51.

The data for this model appear to start near the top of the sine's curve, going down to near its minimum, going back up, and going back down. This feels like one-and-a-half periods of a cosine, so the shift would definitely be to the left. Hence the [imath]+0.02[/imath] (though I would have expected c to have a larger value).

By the way, what did *you* get? What were your steps?

Both example problems gave negative "c" answers, yet one problem has c being added to b and the other is subtracted from b. What is the correct answer here and moving forward, when asked to find sine regression functions on the calculator, if the value of "c" is negative, what would the sine regression function look like?

Either one should be fine. And anyway, you need to become comfortable with "seeing" both patterns as meaning the same thing. It's like in my graphing calculator, when doing linear regressions, I can choose between their [imath]y=a+bx[/imath] and their [imath]y = ax + b[/imath] modelling equations. Why do they have both? I have no idea. Just roll with it.

 

[hivar]

For ex.1, I inputted the day number column into the list for the x-axis with the hours of daylight column on the y-axis. I plotted the data, calculated the sine reg function and I got the same answer H(d)=4.23sin(0.017d - 1.30) + 12.12.

For ex.2, I inputted the time column into the list for the x-axis with the height column on the y-axis. I again plotted the data, found the sine reg function but I wrote h(t) = 3.15sin(1.05d - 0.02) + 7.51 due to "c" being negative and it is wrong.

Can someone explain to me in perhaps simpler terms what I am doing wrong with the value of "c" and why one equation has a negative value of c and a negative in the equation while the other has a negative value of c but it turns that the "c" part of the equation positive?

 

[stapel]

I'm gonna guess there's a typo in the book.

 

[Dr.Peterson]

Both example problems gave negative "c" answers, yet one problem has c being added to b and the other is subtracted from b. What is the correct answer here and moving forward, when asked to find sine regression functions on the calculator, if the value of "c" is negative, what would the sine regression function look like?

I'm gonna guess there's a typo in the book.

It might help if you could show us an image of exactly what the book says, so we could identify a specific error.

 

[hivar]

It might help if you could show us an image of exactly what the book says, so we could identify a specific error.

I attached the screenshot of the Ex.2 answer from the book. They use the same + "c" value in the formula for the follow-up questions as well.

 

[stapel]

I attached the screenshot of the Ex.2 answer from the book.

I note that the solutions posted for Question 21 and Question 23 contain obvious typoes:

[imath]\qquad \textrm{In the function } h(\boldsymbol{\color{mediumblue}{t}}) = 3.15\sin(1.05\boldsymbol{\color{red}{d}} \color{black}{+0.02) + 7.51},[/imath]

[imath]\qquad \textrm{substitute } 45 \textrm{ for } \boldsymbol{\color{mediumblue}{t}}, \textrm{ and solve for } h(\boldsymbol{\color{mediumblue}{t}}).[/imath]

The author of the solution didn't even notice that s/he was using the wrong letter for the variable! This supports the impression that this boils down to a typo.