11 - Applications of Sinusoidal Functions WT PDF

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Applications of Trigonometric Functions We can use sinusoidal functions to model situations. You will have to make graphs to model situations and be able to interpret situations from the graph. Example 1 During high tide, the water depth in a harbour is 22m and during low tide 6 hours later it is 10m. a) Draw a graph to represent the water height for a 24-hour period starting at high tide. b) Find the equation to represent the function. c) What is the height of the water after 3h, 9h and 20h? Solution Graphing Trigonometric Functions Step 1: Start by marking your scale on the graph. Horizontal axis: __________________________________________________ Vertical axis: ____________________________________________________ Step 2: Plot the key points. Where do we want to start? __________________________________________ ________________________________________________________________ How do we find the next points? _______________________________________ ________________________________________________________________ Step 3: Connect the points with a smooth curve.

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Finding the Equation To find the equation we need to know the period, amplitude and if there were any translations. Step 1: Determine the period, then use it to find the k value.

Step 2: Determine the amplitute. Step 3: Determine if there are any translations. Start by deciding if you are going to use the cosine or sine for your function.

Step 4: Write out the equation.

c) Interpreting Information from the Graph Look at the graph and read off the points. The height of the water after 3 hours is _________________ The height of the water after 9 hours is _________________ The height of the water after 20 hours is ________________

Example 2 A pendulum swings back and forth in the same place at a constant speed. The displacement from its

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resting position is described by the function d (t ) = 6 sin t where d is the displacement in centimeters 3 and t is the time is seconds. Draw a graph for the function starting at t = 0 for two cycles. Step 1: Determine the amplitude.

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Step 2: Determine the period.

Step 3: Graph the function using the key points.

When is the pendulum at its resting place?

What is the total distance the pendulum swings through in one cycle?

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Example 3 When you ride a bike, the height of your foot above the road is a circular function of the time you have been pedaling. An equation to model this height is h( t) = 20 − 15 cos2 t, where h is the height in centimeters and t is the time is seconds. a) Graph this equation for two cycles.

b) How far above the ground is your foot at t = 0?

c) What is the radius of the pedal arm?

d) How far above the ground is the center of the circle the pedals make when they go around?

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