# Thread: Trig function word problem

1. ## Trig function word problem

Hi,

I hope someone can help. I'm currently trying to solve this following problem:

These are the characteristics of the functions which I believe are correct (please let me know if I'm wrong):
- amplitude = 16.2
- equation of axis = 1.4
- period = pi/6
- phase shift = 7/2 (assuming that January = 1, and December = 12 on the x-axis)

This results in the equation y=16.2sin(pi/6(x-7/2))+1.2

With this equation, I can now determine when the function is below 0 degree celsius by creating an inequality statement and then solving for x.

Is this the correct way to approach this problem? Please let me know where my thinking is off. Please provide me with hints than just answers! I want to learn. Thank you.

Sincerely,
Olivia

2. ## Re: Trig function word problem

attached is the graph of the function you determined ...

y=16.2sin(pi/6(x-7/2))+1.2
... superimposed on a plot of the temperature data for each month ... what do you think?

3. ## Re: Trig function word problem

It's hard to tell. Do you think the best way to figure out the equation is by plotting the coordinates with graphing technology? I unfortunately don't have a graphing calculator so I'm a bit lost in terms of what I should do. The phase shift is sort of hard to determine without knowing the exact point where it crosses the equation of axis... so I think that some sort of graphing technology would make sense to use. I can't imagine any other way.

4. ## Re: Trig function word problem

Do you think the best way to figure out the equation is by plotting the coordinates with graphing technology?
No, but it helps to check the equation you determined.

I unfortunately don't have a graphing calculator
use an online calculator ...

Graphing Calculator- Free online tool graph functions, finds intersections, table of values. Implicit equations, pan, zoom, & export as image

https://www.desmos.com/calculator

... there are more available, all you have to do is search & choose the one you like.

The phase shift is sort of hard to determine without knowing the exact point where it crosses the equation of axis...
not necessarily ... min Temp in January (month 1) and max Temp in July (month 7) $\implies$ the curve crosses the midline close to month 4 (April) . With a 12 month period, $y = 16.2\sin\bigg[\dfrac{\pi}{6}(x - 4)\bigg] + 1.4$

(second screenshot) not too bad doing it by hand ... the calculator's sine regression program works a bit better (first & third screenshots).

5. ## Re: Trig function word problem

But to say that the midpoint crosses over in April is just an approximation, correct? I say this because the midpoint is at y = 1.4 and April begins at y=2.5... so obviously not exact.

And to solve for when it is below 0 degrees, do you think that I should create an inequality using my equation and then solve accordingly?

6. ## Re: Trig function word problem

I have an update: according to my textbook, the equation to this question is T(t) = 16.2sin(2pi/365(t-116))+1.4

This is pretty exact... and it appears that each month represents approx 30.146 days. So I would need to use a graphing calculator in order to determine where is crosses the equation of axis.