Secondary Mathematics III
Educational Links

Strand: FUNCTIONS - Trigonometric Functions (F.TF)

Extend the domain of trigonometric functions using the unit circle (Standards F.TF.1-3). Model periodic phenomena with trigonometric functions (Standards F.TF.5-7).

• As the Wheel Turns
  In this task, students use trigonometric functions to model the movement of a point around a wheel and, in the case of part (c), through space (F-TF.5). Students also interpret features of graphs in terms of the given real-world context (F-IF.4).
• Bicycle Wheel
  The purpose of this task is to introduce radian measure for angles in a situation where it arises naturally.
• Coordinates of Points on a Circle
  The purpose of this task it to use geometry and algebra in order to understand the behavior of the trigonometric function f(x)=sinx+cosx. The task has been stated in an open ended fashion as there are natural solutions using geometry, or using the trigonometric identity sin2x=2sinxcosx, or algebraically solving a system of equations.
• Equilateral triangles and trigonometric functions
  The purpose of this task is to apply knowledge about triangles to calculate the sine and cosine of 30 and 60 degrees.
• Exploring Sinusoidal Functions
  This task serves as an introduction to the family of sinusoidal functions. It uses a desmos applet to let students explore the effect of changing the parameters in y=Asin(B(x−h))+k on the graph of the function.
• Foxes and Rabbits 2
  The example of rabbits and foxes was introduced to illustrate two functions of time given in a table. We are now in a position to actually model the data given previously with trigonometric functions and investigate the behavior of this predator-prey situation.
• Foxes and Rabbits 3
  The example of rabbits and foxes was introduced to illustrate two functions of time given in a table. The same situation was used in F-TF Foxes and Rabbits 2 to find trigonometric functions modeling the data in the table. The previous situation was somewhat unrealistic since we were able to find functions that fit the data perfectly. In this task, on the other hand, we do some legitimate modelling, in that we come up with functions that approximate the data well, but do not perfectly match, the given data.
• FUNCTIONS - Trigonometric Functions (F.TF) - Sec Math III Core Guide
  The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics III - Trigonometric Functions (F.TF).
• Introduction to the Materials (Math 3)
  Introduction to the Materials in the Mathematics Three of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems.
• Module 6: Modeling Periodic Behavior - Student Edition (Math 3)
  The Mathematics Vision Project, Secondary Math Three Module 6, Modeling Periodic Behavior. In this module students use a Ferris wheel as a context for constructing conceptual understanding of circular trigonometry. They begin by calculating heights on the Ferris wheel, progress to calculating the heights at a given time on the Ferris wheel, and then, graphing the heights to show a sine function.
• Module 6: Modeling Periodic Behavior - Teacher Edition (Math 3)
  The Mathematics Vision Project, Secondary Math Three Module 6, Modeling Periodic Behavior. In this module students use a Ferris wheel as a context for constructing conceptual understanding of circular trigonometry. They begin by calculating heights on the Ferris wheel, progress to calculating the heights at a given time on the Ferris wheel, and then, graphing the heights to show a sine function.
• Module 7: Trig. Functions, Equations & Identities - Student Edition (Math 3)
  The Mathematics Vision Project, Secondary Math Three Module 7, Trigonometric Functions, Equations, and Identities, students work with more trigonometric graphs, beginning with the familiar Ferris wheel context.
• Module 7: Trig. Functions, Equations & Identities - Teacher Edition (Math 3)
  The Mathematics Vision Project, Secondary Math Three Module 7, Trigonometric Functions, Equations, and Identities, students work with more trigonometric graphs, beginning with the familiar Ferris wheel context.
• Properties of Trigonometric Functions
  The goal of this task is to use the unit circle and rigid transformations in order to establish some fundamental trigonometric function identities.
• Representing Trigonometric Functions - Ferris Wheel
  This lesson unit is intended to help educators assess how well students are able to model a periodic situation, the height of a person on a Ferris wheel, using trigonometric functions, and interpret the constants a, b, c in the formula h = a + b cos ct in terms of the physical situation, where h is the height of the person above the ground and t is the elapsed time.
• Special Triangles 1
  Using known facts about the unit circle and isosceles triangles together with the Pythagorean Theorem, we can derive the sine and cosine of special angles, in this case of π/4. This task can be done as a mini lecture soliciting responses from the students, or as a challenge problem for students to ponder and discuss.
• Special Triangles 2
  Using known facts about the unit circle and isosceles triangles together with the Pythagorean Theorem, we can derive the sine and cosine of special angels, in this case of π/6. This task can be done as a mini lecture soliciting responses from the students, or as a challenge problem for students to ponder and discuss.
• Trig Functions and the Unit Circle
  The purpose of this task is to help students make the connection between the graphs of sint and cost and the x and y coordinates of points moving around the unit circle. Students have to match coordinates of points on the graph with coordinates and angles in the diagram of the unit circle.
• Trigonometric functions for arbitrary angles
  The purpose of this task is to examine trigonometric functions for obtuse angles. The values sinx and cosx are defined for acute angles by referring to a right triangle one of whose acute angles measures x. For an obtuse angle, no such triangle exists and so an alternate definition is required. Prior to working on this task, students should have experience working with trigonometric functions and how they relate to the unit circle.
• Trigonometric Identities and Rigid Motions
  The purpose of this task is to apply translations and reflections to the graphs of the equations f(x)=cosx and g(x)=sinx in order to derive some trigonometric identities.
• What exactly is a radian?
  Radians are often mysterious to students, yet they are a very straight forward way to measure an angle by relating the measure of the angle to the length of the arc on the unit circle it subtends. This task is not designed to discover the definition of radian, rather it allows students to make meaning out of the definition.

---

http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey  Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer  Throndsen.

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.