In this lesson, students are building skills that will help them in mathematical modeling (MP4). They learn to use the unit circle as a starting point for thinking about circular motion and then scale their results appropriately to model clocks of different sizes or scale and translate them to model a Ferris wheel whose center is not at \((0,0)\). Students find the coordinates of points on these circles algebraically and then consider what they represent in context. They also consider the difference between approximating the coordinates and representing them exactly as expressions using cosine and sine. These ideas are developed further throughout the unit, leading eventually to the notions of amplitude and midline of trigonometric functions and using these functions for modeling different types of periodic relationships.
Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems (MP5) and we recommend making technology available.
- Use sine and cosine to find coordinates of points on a circle in all 4 quadrants.
- Let’s find coordinates on a circle.
It will be helpful for students to have access to scientific calculators if they choose to compute approximate trigonometric values to help give meaning to their answers in context.
It may be helpful for some students to have extra copies of the gridded clock faces in the activity “Clock Coordinates.” Consider printing and making extras available using the Blackline Master.
Devices are required for the digital version of the activity “Around a Ferris Wheel.”
- I can use cosine and sine to figure out information about points rotating in circles.
The identity \(\sin^2(x) + \cos^2(x) = 1\) relating the sine and cosine of a number. It is called the Pythagorean identity because it follows from the Pythagorean theorem.
The length of an interval at which a periodic function repeats. A function \(f\) has a period, \(p\), if \(f(x+p) = f(x)\) for all inputs \(x\).
A function whose values repeat at regular intervals. If \(f\) is a periodic function then there is a number \(p\), called the period, so that \(f(x + p) = f(x)\) for all inputs \(x\).
The circle in the coordinate plane with radius 1 and center the origin.