Wayne RESA
Curriculum, OS/MAISA 

Common Core Initiative

...
Overarching Questions and Enduring Understandings

How can the unit circle be used to develop a circular definition of trigonometric functions?

...
Graphic Organizer
...
Unit Abstract

Previous studies of trigonometry have included right triangle trigonometric relationships and the Laws of Sines and Cosines. In this unit students develop a circular definition of trigonometric functions using both degree and radian measure. The unit circle is the basis for these definitions and leads to graphs of the sine and cosine curves. Students can then define the tangent and inverse trigonometric functions based on the sine and cosine. Making connections between the other trigonometric functions and the graphs and tables of the sine and cosine functions provides a context for students to develop the graphs and tables of the tangent, cotangent, secant, and cosecant functions. Building on their previous knowledge of transformations (both algebraic and geometric), students explore and graph the effects of transformations (amplitude, period, and phase shift) on the sine and cosine curves.

...
Expectations/Standards
MI: Mathematics
MI: HS: Functions
Building Functions
HSF-BF.B. Build new functions from existing functions.
HSF-BF.B.4. Find inverse functions.
HSF-BF.B.4a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2x³ for x > 0 or f(x) = (x+1)/(x–1) for x ? 1.
HSF-BF.B.4c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
HSF-BF.B.4d. (+) Produce an invertible function from a non-invertible function by restricting the domain.
Trigonometric Functions
HSF-TF.A. Extend the domain of trigonometric functions using the unit circle.
HSF-TF.A.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
HSF-TF.A.2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
HSF-TF.A.3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for p/3, p/4 and p/6, and use the unit circle to express the values of sine, cosines, and tangent for x, p + x, and 2p – x in terms of their values for x, where x is any real number.
HSF-TF.A.4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
HSF-TF.B. Model periodic phenomena with trigonometric functions.
HSF-TF.B.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
HSF-TF.B.6. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
HSF-TF.B.7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
HSF-TF.C. Prove and apply trigonometric identities.
HSF-TF.C.8. Prove the Pythagorean identity sin²(?) + cos²(?) = 1 and use it to calculate trigonometric ratios.
HSF-TF.C.9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
Unit Level Standards

There are standards listed in this section for two reasons.or a function that models a relationship between two quantities, interpret key features of graphs a

  1. The standards have been modified to be appropriate for this unit. Text in gray font is part of the Michigan K-12 standard but does not apply to this unit. Text in brackets denotes a modification that has been made to the standard.
  2. The standards contain content that is developed and/or utilized across multiple units.

Modified For this Unit

HSF-IF.C. Analyze functions using different representations.

HSF-IF.C.7e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

 

Building Functions

HSF-BF.A. Build a function that models a relationship between two quantities.

HSF-BF.A.1a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

 

Developed and/or utilized across multiple units

Quantity

HSN-Q.A. Reason quantitatively and use units to solve problems.

HSN-Q.A.2. Define appropriate quantities for the purpose of descriptive modeling.

HSN-Q.A.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

 

Interpreting Functions

HSF-IF.A. Understand the concept of a function and use function notation.

HSF-IF.A.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

HSF-IF.A.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

 

Interpreting Functions

HSF-IF.B. Interpret functions that arise in applications in terms of the context.

HSF-IF.B.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★

HSF-IF.B.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

 

HSF-IF.C. Analyze functions using different representations.

HSF-IF.C.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

 

Building Functions

HSF-BF.A. Build a function that models a relationship between two quantities.

HSF-BF.A.1. Write a function that describes a relationship between two quantities.

HSF-BF.A.1b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

 

HSF-BF.B. Build new functions from existing functions.

HSF-BF.B.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

...
Essential Questions
Essential/Focus Questions
  1. What is the relationship between degree and radian measures of angles? Why or when would you use degree or radian?
  2. How can the effect of transformations on the sine and cosine curves be seen in the graphs and tables of these functions?
  3. How can the unit circle be used to generate the sine and cosine graphs?
  4. Why are the trigonometric functions periodic?
  5. How do you know if a function is periodic?
Content (Key Concepts)

amplitude

asymptote

cosecant

cosine

cotangent

degree and radian relationship and conversion

domain

maxima

minima

period

phase shift

range

relate graphs of trigonometric functions to their inverses

secant

sine

tangent

transformations of trig function from the parent functions

unit circle

...
Unit Assessment Tasks
Skills (Intellectual Processes)

Standards for Mathematical Practice

Students will have opportunities to:

  • look for and express regularity in repeated reasoning observing the relationships between the sine and cosine and using the unit circle to generate the periodic graphs of these functions;
  • model with mathematics using trigonometric functions and transformations of these functions to model periodic behavior; and
  • use appropriate tools strategically such as graphing calculators or computer applets to analyze the sine and cosine functions and changes in their characteristics as parameter changes are made and to find values of trigonometric functions for specific angle measures.

 

 

...
Lesson Plan Sequence
Lesson Plans (Sequence)
Resources
...
Wayne RESA