Sine Function for Sunset Times (Check Answer)
I want to check the answer I got on the following question, I've attached the graph for reference to help with the question.
73. Use the following graph to:
(I've put the x-values below the image), and the y-values go up by 2.
90, 180, 270, 360, 450, 540, 630, 270
a) Determine an equation that models the time of the sunset in Saskatoon over 720 days. Justify your answer.
The graph is in the form y= asin(2π/p)(x-d) + c
a is amplitude which is the distance between min value (16) and max value (20) divided by 2.
Graph begins upward so a is positive, so a=2.
p is period which is the time it takes to complete one cycle, which is 360 days. So p= 2π/360 or simplified to p=π/180
(The value of p is what I am concerned about, am I writing it right or is it incorrect?)
d is the phase shift or horizontal shift on the x-axis from x=0. We can see the x-value is resting at 0, so d=0
c is the vertical shift of the position on y-axis or in other words, the y-value when the pattern begins, or y-value when x=0. The time of sunset begins up at 18h,
so c= 18
Therefore the equation is y= 2sin (2π/360)(x) + 18 , simplified to y=2sin (π/180)(x) + 18.
When I check the equation to solve for a known value, such as y=20h when x= 90 days.
I get y=2sin (π/180)(90) + 18
Which is close, but is still a one hour difference. So according to the equation y=21 and not y=20.
Or is it okay to have that much of a difference in calculation since we're using a real life application where the results are not always exact??
b) State range of function
(Y E R / 16 < Y < 20}
Can you guys just confirm?? Thanks!!
Re: Sine Function for Sunset Times (Check Answer)
The last part you got was wrong: the answer is exactly 20 and not 21.14.
Note that you are calculating sine in radians and not degrees. In radians sin(pi/2) is exactly 1.
If you are using a calculator, make sure the setting is for radians and not degrees (or something else).