The latest sunrise in Montreal was on Dec. 22 at 9:15 AM . According to the almanac, the earliest sunrise occurred on the 22nd of June at 3:15 AM . The sunrise times on other dates can be predicted using a sinusoidal equation.
**Assume there is no daylight savings time in Montreal.**
THE QUESTION: Part I
(a) Write two equations for the described function above; one using cosine and the other using sine.
THE SOLUTION: Part I
Read the given info and convert it into information that can be used in an equation. For example, the time is not going to be written 9:15 on the graph but 9.25 because 15 min. is a quarter of an hour.
Make two lists for the parameters A, B, C & D. One set will be used for the cosine equation and the other will be used for the sine equation.
Find the parameters in the mean of DABC [stretches before translations]. To find parameter D, add the min. and max. value and then divide by 2 to find the sinusoidal axis .
To find parameter A, subtract the average value from the maximum value to get A.
cont‘d on the next slide; Hupsha, hupsha now…
Parameter B is equal to 2 π divided by the period, which happens to be the number of days in a year; 364.
The phase shift (C) is found depending on what kind of equation is being used. If the cosine equation is being found, the maximum value is usually on the y-axis. But the information tells us that the maximum value occurs on Dec. 22, 9 days before Jan. 1 [the y-axis].
The phase shift in the sine equation is determined by finding out the distance of the average value to the y-axis.
Finally, to get the equations, plug in the values found into the general formula.
THE QUESTION: Part II
(b) Sketch the graph for the sinusoidal function described in the problem.
THE SOLUTION: Part II
Acknowledge that Dec. 22 is 9 days before Jan. 1. [use this later in graphing stage.]
To find the values within the period of one cycle in the graph, subtract nine days from the full, half, and quarter periods.
On the y-axis, label where the min. value, max. value and sinusoidal axis are. Plot the points according to info; max value at Dec. 22.
Label the axes.
THE QUESTION: Part III
(c) Use one of the equations in (a) to predict the time of sunrise on September 7.
THE SOLUTION: Part III
Find out what the day of the year September 7 th is by adding up the total number of days in each month up to the given date, assuming there is no daylight savings time.
Since d lies along the x-axis, treat the number of days as an x-coordinate and plug in as d in the formula, either sine or cosine and solve.
Sun, sun...Mr. Golden Sun.
THE QUESTION: Part IV
(d) What is the average sunrise time throughout the year?
THE SOLUTION: Part IV
The avg. sunrise time equals the sinusoidal axis. To find this, find parameter D by adding the minimum and maximum value and divide by two.
Looking at the sun gives me a feeling of freedom. Too bad Max can’t feel this way…
Sunrise Over Rocks, Lighthouse Beach by flickr user Captain Capture