Parametric Equations (With Trig) (Lesson 5.8 Day 2)
Unit 5 - Day 15
Graph parametric equations involving trigonometry using tables
Use the Pythagorean identity to convert between parametric and Cartesian equations of circles and ellipses
Understand the advantages of parameterizing a curve
Quick Lesson Plan
In today’s activity, students use parametric equations to track Jack’s position on a Ferris wheel, realizing that his vertical and horizontal position can both be described using trigonometric functions. In questions 1-2, students evaluate and solve parametric equations. In question 4 students graph the parametric equations by first making a table, and in question 5, students convert to Cartesian form, reviewing fundamental ideas about equations of circles and the Pythagorean theorem. Question 3 is critical to understanding these relationships. The fact that Jack’s position is always 50 meters from the center of the wheel provides an equation relating his vertical and horizontal position at any time during the ride.
Note that his horizontal position includes the sine function and his vertical position includes the cosine function. This may throw students for a loop (no pun intended!) so check early on that students are evaluating the functions and filling out the table correctly. It is worthwhile for students to understand that t=0 is a time, and doesn’t refer to a particular position on the unit circle. Furthermore Jack’s horizontal and vertical position is not simply the x and y coordinate on the unit circle at a given angle.
Today’s lesson connects many important ideas from Unit 4 and Unit 5. Right triangle trigonometry, equations of circles, and even polar coordinates are at play here! The use of the Pythagorean identity is critical for converting parametric equations with trig in them to Cartesian equations. Students should not try to eliminate the parameter by solving for t like they did in yesterday’s lesson.
You may wish to spend additional time discussing the Pythagorean identity, proving it not just using the unit circle, but any circle of radius r. Using the polar conversion equations x=rcosθ and y=rsinθ and plugging these into the circle equation x^2+y^2=r^2 is another way of showing this important property. This identity becomes especially important when converting equations for ellipses where students must solve for cos t and sin t in their equations.
One of the learning targets for today is to discuss the possible advantages of parameterizing a curve. Two main ideas should surface from this discussion. First, parameterized curves help show the direction in which an object moved (the order in which the points were graphed). Second, graphs that do not represent functions (like circles and ellipses) can be described by parametric equations that are functions. This is helpful since much of the mathematics that students have seen in school is tied to functions, not relations.