6.6 Modelling with Trigonometric Functions
Chapter
Chapter 6
Section
6.6
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Solutions 16 Videos

A cosine curve has an amplitude of 3 units and a period of 3\pi radians. The equation of the axis is y = 2, and a horizontal shift of \frac{\pi}{4} radians to the left has been applied. Write the equation of this function.

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1.44mins
Q1

Determine the value of the function \displaystyle y=3\cos\left(\dfrac{2}{3}\left(x + \dfrac{\pi}{4}\right)\right) + 2 if x = \frac{\pi}{2}, \frac{3\pi}{4}, and \frac{11\pi}{6}.

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2.38mins
Q2

A cosine curve has an amplitude of 3 units and a period of 3\pi radians. The equation of the axis is y = 2, and a horizontal shift of \frac{\pi}{4} radians to the left has been applied.

Sketch a graph of the function in described above. Use your graph to estimate the x-value(s) in the domain 0 < x < 2\pi, where y = 2.5, to one decimal place.

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2.02mins
Q3

The height of a patch on a bicycle tire above the ground, as a function of time, is modelled by one sinusoidal function. The height of the patch above the ground, as a function of the total distance it has travelled, is modelled by another sinusoidal function. Which of the following characteristics do the two sinusoidal functions share: amplitude, period, equation of the axis?

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1.44mins
Q4

Mike is waving a sparkler in a circular motion at a constant speed. The tip of the sparkler is moving in a plane that is perpendicular to the ground. The height of the tip of the sparkler above the ground, as a function of time, can be modelled by a sinusoidal function. At t = 0, the sparkler is at its highest point above the ground.

  • What does the amplitude of the sinusoidal function represent in this situation?
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1.11mins
Q5a

Mike is waving a sparkler in a circular motion at a constant speed. The tip of the sparkler is moving in a plane that is perpendicular to the ground. The height of the tip of the sparkler above the ground, as a function of time, can be modelled by a sinusoidal function. At t = 0, the sparkler is at its highest point above the ground.

  • What does the period of the sinusoidal function represent in this situation?
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0.23mins
Q5b

Mike is waving a sparkler in a circular motion at a constant speed. The tip of the sparkler is moving in a plane that is perpendicular to the ground. The height of the tip of the sparkler above the ground, as a function of time, can be modelled by a sinusoidal function. At t = 0, the sparkler is at its highest point above the ground.

  • What does the equation of the axis of the sinusoidal function represent in this situation?
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0.41mins
Q5c

Mike is waving a sparkler in a circular motion at a constant speed. The tip of the sparkler is moving in a plane that is perpendicular to the ground. The height of the tip of the sparkler above the ground, as a function of time, can be modelled by a sinusoidal function. At t = 0, the sparkler is at its highest point above the ground.

  • If no horizontal translations are required to model this situation, should a sine or cosine function be used?
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0.41mins
Q5d

To test the resistance of a new product to temperature changes, the A product is placed in a controlled environment. The temperature in this environment, as a function of time, can be described by a sine function. The maximum temperature is 120^{o}C, the minimum temperature is 260^{o}C, and the temperature at t = 0 is 30^{o}C. It takes 12 h for the temperature to change from the maximum to the minimum. If the temperature is initially increasing, what is the equation of the sine function that describes the temperature in this environment?

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3.29mins
Q6

A person who was listening to a siren reported that the frequency of the sound fluctuated with time, measured in seconds. The minimum frequency that the person heard was 500 Hz, and the maximum frequency was 1000 Hz. The maximum frequency occurred at t = 0 and t = 15. The person also reported that, in 15, she heard the maximum frequency 6 times (including the times at t = 0 and t = 15). What is the equation of the cosine function that describes the frequency of this siren?

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2.01mins
Q7

A contestant on a game show spins a wheel that is located on a plane perpendicular to the floor. He grabs the only red peg on the circumference of the wheel, which is 1.5 m above the floor, and pushes it downward. The red peg reaches a minimum height of 0.25 m above the floor and a maximum height of 2.75 m above the floor. Sketch two cycles of the graph that represents the height of the red peg above the floor, as a function of the total distance it moved. Then determine the equation of the sine function that describes the graph.

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3.02mins
Q8

At one time, Maple Leaf Village (which no longer exists) had North America?s largest Ferris wheel. The Ferris wheel had a diameter of 56 m, and one revolution took 2.5 min to complete. Riders could see Niagara Falls if they were higher than 50 m above the ground. Sketch three cycles of a graph that represents the height of a rider above the ground, as a function of time, if the rider gets on at a height of 0.5 m at t = 0 min. Then determine the time intervals when the rider could see Niagara Falls.

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7.33mins
Q9

The number of hours of daylight in Vancouver can be modelled by a sinusoidal function of time, in days. The longest day of the year is June 21, with 15.7 h of daylight. The shortest day of the year is December 21, with 8.3 h of daylight.

  • Find an equation for n(t), the number of hours of daylight on the nth day of the year.
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3.42mins
Q10a

The number of hours of daylight in Vancouver can be modelled by a sinusoidal function of time, in days. The longest day of the year is June 21, with 15.7 h of daylight. The shortest day of the year is December 21, with 8.3 h of daylight.

(b) Use your equation to predict the number of hours of daylight in Vancouver on January 30th.

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0.23mins
Q10b

A nail is stuck in the tire of a car. If a student wanted to graph a sine function to model the height of the nail above the ground during a trip from London, Ontario, to Waterloo, Ontario, should the student graph the distance of the nail above the ground as a function of time or as a function of the total distance travelled by the nail? Explain your reasoning.

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0.57mins
Q12

A clock is hanging on a wall, with the centre of the clock 3 m above the floor. Both the minute hand and the second hand are 15 cm long. The hour hand is 8 cm long. For each hand, determine the equation of the cosine function that describes the distance of the tip of the hand above the floor as a function of time. Assume that the time, t, is in minutes and that the distance, D(t), is in centimetres. Also assume that t = 0 is midnight.

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2.06mins
Q13
Lectures 4 Videos

Ex A Ferris Wheel spins once every 12 secs. The ferris wheel is 2 m from the ground and has a radius of 6m. If the rider gets on from the lowest point of the ferris wheel, how high is the rider after 20 secs?

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5.17mins
Ferris Wheel Problem ex1