Unit 8: Modeling Periodic Behavior
Lesson Videos
Focus Standards
Learning Focus
Additional Resources
A Develop Understanding Task
Uses reference triangles, right triangle trigonometry, and the symmetry of a circle to find the y -coordinates of points on a circular path. Introduces the idea that right triangle trigonometric ratios can be used as a tool to describe periodic behavior.
Apply right triangle trigonometry to a circular context.
How can right triangle trigonometry be applied to find how far points on a circle are away from a horizontal diameter of the circle?
A Solidify Understanding Task
Extends strategies from the previous lesson to find the y-coordinates of points on a circular path at various instances in time. Introduces circular trigonometric functions.
Write a trigonometric function to model a context.
How can I determine the vertical height of a rider on a moving Ferris wheel?
A Solidify Understanding Task
Extends the definition of sine from a right triangle trigonometric ratio to a function of an angle of rotation by building on the work from the previous lessons.
Extend the definition of sine to include all angles of rotation.
How can we define the sine function for angles larger than 90°?
A Solidify Understanding Task
Uses the graph of a sine function to model circular motion and relates features of the graph to the parameters of the function.
Graph sine functions of the form .
How can I represent the vertical motion of a rider on a Ferris wheel graphically?
How does changing the speed, height, or radius of the Ferris wheel affect the graph and the function equation?
A Practice Understanding Task
Extends the definition of the cosine from a right triangle trigonometric ratio to a function of an angle of rotation using the same context and strategies from previous lessons.
Locate points in a plane using coordinates based on horizontal and vertical movements or based on circles and angles.
Use degrees and radians to measure angles.
Are there other ways to describe the location of a point in the plane other than by giving its - and -coordinates?
What proportionality relationships can I find between corresponding points and arc lengths of concentric circles? How can I justify why those proportionality relationships exist?
A Develop Understanding Task
Introduces radians as a unit for measuring angles on concentric circles. Develops understanding of the relationship between arc length measurements and radian angle measurements.
Locate points in a plane using coordinates based on horizontal and vertical movements or based on circles and angles.
Use degrees and radians to measure angles.
Are there other ways to describe the location of a point in the plane other than by giving its x - and y-coordinates?
What proportionality relationships can I find between corresponding points and arc lengths of concentric circles? How can I justify why those proportionality relationships exist?
A Solidify Understanding Task
Solidifies understanding of radians and uses the proportionality relationship of radian measure to locate points on concentric circles.
Calculate arc length for angles of rotation measured in radians.
Visualize the size of angles measured in radians, including radians given in decimal form.
How large is a radian? How can I estimate the size of angles measured in radians relative to degrees?
What good are radians? Are any calculations made easier when the angle is measured in radians? Are any contexts easier to describe using radians?
A Solidify Understanding Task
Redefines radian measure of an angle as the length of the intercepted arc on a unit circle. Provides opportunity to begin creating a unit circle with sine and cosine values.
Find a relationship between the arc length and ( x , y ) coordinates of a point on the circle of radius 1 .
How does the unit circle make our work with trigonometric functions easier?
A Practice Understanding Task
Defines sine and cosine on the unit circle in terms of angles of rotation measured in radians.
Apply special right triangles to the unit circle.
Are there any angles for which I can find the value of the sine or cosine without using a calculator?