Question

One January 1st at West Bay, the firsy low tide was at 1:30am and at 0.7m, and the first high tide was at 7:45am at 2.8m. Assuming high tides and low tides occur at regular intervals.

1. Find two equivalent trigonometric equations that model the height in m of the tide at West Bay in terms of tike (t) in hours since midnight. (You must use two different trigonometric functions from sine, cos or tan, and set t=0 as midnight on 1st January for each question).
2. Justify and prove that the two equations are equivalent, unless the proof is shown in your working.

Answer 1

To model the height of the tide at West Bay, you can use the sine or cosine function with the appropriate parameters. Both equations are equivalent and can represent the same periodic motion.

Step-by-step explanation:

To model the height of the tide at West Bay in terms of time (t) since midnight on January 1st, we can use trigonometric functions. One equivalent equation can be represented by the sine function, where the height (h) is given by h = A * sin(B(t - C)) + D. Another equivalent equation can be represented by the cosine function, where the height (h) is given by h = A * cos(B(t - C)) + D. In both equations, A represents the amplitude, B represents the frequency (the number of cycles that occur within a given time period 2π), C represents the phase shift (the time it takes for the initial cycle to start), and D represents the vertical shift, which is the average height of the tide.

These two equations are equivalent because they belong to the same family of trigonometric functions and can describe the same periodic motion. The sine and cosine functions are related through a phase shift of π/2 or 90 degrees. So, when you shift one equation by π/2, it becomes equivalent to the other equation. Therefore, both equations can be used to represent the height of the tide at West Bay in terms of time since midnight.