Trigonometric Functions

Question

Throughout any given month, the maximum and minimum ocean tides follow a periodic pattern. Last year, at a certain location on the California coast, researchers recorded the height of low tide, with respect to sea level, each day during the month of July. The lowest low tide was first measured on July 11, at -1.4 feet. The highest low tide was first measured on July 4, at 1.8 feet. The average low tide for the month of July was measured to be 0.2 feet.

Part A

Which curve would you choose to model this function, sine or cosine? Give your reasons.

Part B

Find the amplitude of the function. Explain its meaning in the context.

Part E

Find the period of the function. Explain its meaning in the context.

Part C

Find the vertical shift of the function. Explain its meaning in the context.

Part D

Find the phase shift of the function. Explain its meaning in context.

Part E

Write the function for the curve from all of the key features you just found.

Part F

What are the days of the month when the low tide is projected to be the average height? What do you suppose a decimal value for a day in the month means?

Part G

The actual low tide height recorded on a given day could vary from the function created as a model. Give some reasons for why you think this happens.

Answer 1

Part A: I would pick a cosine wave since the function starts first at the max (July 4 is the highest low tide) and cosine starts at the max. But it really isn’t critical, just requires a different phase shift.

Part B: The amplitude is the max - average so 1.8 - 0.2 or 1.6 feet. The meaning is how far the tide varies from the average.

Part C: The vertical shift is the average - zero so 0.2 - 0 or 0.2 feet. The meaning is that the average low tide is 0.2 feet above sea level.

Part D: The phase shift is 4 days. That’s the time delay of this wave from the standard cosine wave. If we did not apply a phase shift the maximum would occur at time t = 0 or the first day. No times are given so we assume the time of day is unimportant in this context (which, by the way, is probably untrue)

Part E: The function h(t) where “h” represents the height of the low tide and “t” represents the time in days beginning July 1 is: h(t) = 1.6cos(pi/7(t - 4)) + 0.2

Part F: Sketching out the curve, we see that the cosine wave crosses the “average” 3.5 days prior to its max on July 4 and then it would be “average” on day 0.5, day 7.5, day 14.5, day 21.5, and day 28.5 for a total of 5 times. Decimal aliens would just represent portions of a day.

Part G: Variations could occur due to other gravitational factors (alignment or misalignment of the sun & moon or other planets) or local geographic features (bays, inlets, etc).