MCR3U - Word Problems for Sine and Cosine Functions

Mr. Ray introduces a Grade 11 university math lesson focusing on trigonometric functions, specifically sine and cosine, in the context of word problems. These problems involve periodic behaviors such as the motion of a Ferris wheel, tide fluctuations, piston movements, or a pendulum's swing. He emphasizes the importance of sketching the situation, understanding maximum and minimum values, amplitude, and period to model these behaviors using sine or cosine functions. The lesson aims to guide students through the process of creating equations to represent these scenarios and using them for predictions.

The script delves into a detailed example of a Ferris wheel with a radius of 9.5 meters, rotating every 10 seconds, and positioned 1.2 meters above the ground. Mr. Ray explains how to sketch the Ferris wheel and its corresponding graph, taking into account the period, amplitude, and vertical scale relative to the ground. He demonstrates how to determine the maximum, minimum, and center values for the rider's height above the ground. The process includes understanding the starting point of the ride and how it affects the graph and the equation that will be used to model the situation.

Mr. Ray continues the Ferris wheel example by constructing the sine function equation that models the rider's height above the ground as a function of time. He identifies the amplitude as 9.5 meters, the period as 10 seconds, and calculates the necessary values for the equation, including the phase shift. The chosen base function is sine, and adjustments are made to account for the rider starting at the bottom of the wheel. The final equation is used to predict the rider's height at specific times and to explore how changes in the Ferris wheel's characteristics would affect the equation.

The script discusses how different changes to the Ferris wheel scenario would impact the sine function equation. These changes include the wheel turning more slowly, increasing the size of the wheel, raising the wheel off the ground, and altering the rider's starting position. Mr. Ray explains how each modification affects the period, amplitude, vertical translation, and phase shift parameters of the equation, reinforcing the understanding of how the equation represents real-world changes.

Mr. Ray instructs on sketching the graph to represent two complete cycles of the Ferris wheel over a 20-second period. He provides a step-by-step guide to plotting the key points of the graph, such as the maximum and minimum heights and the center value, over time. The graph is an essential tool for visualizing the periodic nature of the Ferris wheel's rotation and serves as a foundation for applying the sine function equation.

The script describes how to use the constructed sine function equation to find the rider's height above the ground after seven seconds. Mr. Ray demonstrates the process of substituting the time variable into the equation and evaluating it step by step. He highlights the importance of showing the calculation process and rounding the final answer to an appropriate number of decimal places for accuracy.

Mr. Ray tackles a more complex question: finding the times when the rider's height above the ground is 10 meters. He uses the previously derived equation and manipulates it to solve for the time variable, illustrating a methodical approach to reverse the operations within the equation. The process involves isolating the time variable and using inverse sine to find the specific moments when the rider's height matches the given value.

The lesson shifts to a new example involving tide fluctuations in the Bay of Fundy, where the water level rises and falls dramatically due to tidal forces. Mr. Ray outlines the parameters for the sine function that models these changes, including the amplitude, period, and average sea level. He constructs the equation for the sine function that represents the depth of the water over a 24-hour period, starting at midnight with an average sea level.

The script addresses a variation of the tide problem where the water is at an average sea level at 2 AM instead of midnight. Mr. Ray explains how to adjust the previously derived sine function equation to account for this change in the starting time. He introduces a phase shift to the equation to reflect the delayed start of the tide coming in, demonstrating the flexibility of the model to adapt to different scenarios.

In conclusion, Mr. Ray summarizes the lesson and encourages students to practice solving word problems involving trigonometric functions. He offers additional handouts for practice, including problems on tides and Ferris wheels, to help students gain confidence and proficiency in applying these mathematical concepts to real-world situations.