TRIG WHEEL 5 the Cosine waveform
A Ferris wheel reaches a maximum height of 30 m. The lowest point of the wheel is 2 m above ground. The wheel travels one complete revolution in 1 minute (60 seconds). Model the wheel using a negative cosine wave function. This time however, the passenger boards the Ferris wheel at its lowest point. As before the function has amplitude (A), frequency (B), horizontal shift (C) and vertical shift (D). You should practise drawing this wave form before attempting any of these questions. This includes understanding how to express the cosine waveform as a sine wave.
Draw a diagram to help you with the problem.
Draw a diagram to help you with the problem.
amplitude (A)
The amplitude is the radius of the circle or max - min divided by 2, which is (30 - 2)/2
= 14 m.
= 14 m.
Frequency (B)
The frequency B is equal to the distance of 2π divided by the time for one period, which is:
horizontal shift (C)
The cosine wave that you have drawn is the same as a sine wave that has been translated to the right on the x axis. It has moved by 1/4 or π/2. In this example that distance along the x axis happens to be 15 seconds because 1/4 x 60 = 15. So the value of C = 15. It is written as (t - 15)
Vertical shift (D)
Because the wheel has been lifted or 'shifted' vertically, the axle (circle origin) is now at a height of (max + min)/2 = (30 + 2)/2 = 16 m. So D = 16 and the equation for the function becomes.
on the calculator
setting up the display on the calculator
Sketching the graph will help with setting up the calculator screen. First go into graphing mode. Learn how your version of calculator works because there are some slight differences between models, especially if you have the old green and white colour scheme (right).
Enter in the equation in this manner (or a method you were taught previously that works for you) notice the use of brackets. |
The x and t mean the same thing on the calculator. Now go into view window to adjust the settings so as to get a reasonable image on the screen.
Both calculators display the same graph. This shows that cosine function can be expressed as a sine function by shifting the sine wave by 1/4 period. Here 1/4 of 60 seconds is 15 seconds. The value of D, the vertical shift is not visible on the displays as they are off screen. They are '+ 16'