Unit 2
Trigonometric Functions

 

Algebra II

 

Unit Topic and Length:

 

Building on their previous work with functions, and on their work with trigonometric ratios and circles in Geometry, in Unit 2 students will now use the coordinate plane to extend trigonometry to model periodic phenomena. In Geometry students will have used basic trigonometric ratios to solve problems involving right triangles. This unit will be the first introduction to the concept of a radian as an angle measure. Students will understand the radian measure of an angle as the length of the arc on the unit circle subtended by the angle. Students will understand the unit circle and its usefulness to extend trigonometric functions to all real numbers. Additionally, students will prove the Pythagorean identity sin2(θ) + cos2(θ) = 1, and use it in their work with angles, measures, and location. Work in this unit will prepare students for more extensive graphing, interpreting, and modeling of trigonometric functions, along with other functions, in Unit 3.

Standards for Mathematical Practice

MP.1 Make sense of problems and persevere in solving them.

MP.2 Reason abstractly and quantitatively.

MP.3 Construct viable arguments and critique the reasoning of others.

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

MP.8 Look for and express regularity in repeated reasoning.

Louisiana Student Standards for Mathematics (LSSM)

F-TF: Trigonometric Functions

A. Extend the domain of trigonometric functions using the unit circle.

F-TF.A.1

 

 

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

*I can define a radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

*I can define terminal and initial side of an angle on the unit circle.

*I can 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.

 

F-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.

*I can explain the relationship between the unit circle and the coordinate plane.

B. Model periodic phenomena with trigonometric functions.

F-TF.B.5

 

 

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

*I can define and recognize the parameters of trigonometric functions.

*I can interpret trigonometric functions in real-world situations.

*I can identify and model periodic phenomena in real-world situations.

*I can choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

C. Prove and apply trigonometric identities.

F-TF.C.8

 

 

Prove the Pythagorean identity and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

*I can define trigonometric ratios as related to the unit circle.

*I can prove the Pythagorean identity .

*I can use the Pythagorean identity, , to find , , or , given , , or , and the quadrant of the angle.

 

F-BF: Building Functions

B. Build new functions from existing functions

F-BF.B.3

 

 

Identify the effect on the graph of replacing by , , , and for specific values of (both positive and negative); find the value of 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.

*I can perform transformations on functions now including sinusoidal functions. (Extend skills from Unit 1).

S-ID: Statistics and Probability

Interpreting Categorical and Quantitative Data

B. Summarize, represent, and interpret data on two categorical and quantitative variables.

S-ID.B.6

 

 

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context.

*I can categorize data as cyclical (sinusoidal) or not.

*I can create a function to model real world data and use the function to predict values.

 

Additional Standards for Honors Classes

 

(+) F-TF.A.3 Use special triangles to determine geometrically the values of sine, cosine, tangent for , and use the unit circle to express the values of sine, cosine, and tangent for , , and in terms of their values for , where is any real number.

 

(+) F-TF.C.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. (sum and difference formulas)

Enduring Understandings:

 

*A radian measure of an angle is the length of the arc on the unit circle subtended by the angle.

*The unit circle enables the extension of the domain of trigonometric functions to include all real numbers.

*Trigonometric functions can be used to model periodic phenomena.

*The Pythagorean identity is very useful when finding sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

 

Essential Questions:

 

*How can you find the measure of an angle in radians?

*What is the unit circle?

*How can you use the unit circle to define the trigonometric functions of an angle?

*What are the characteristics of the real-life problems that can be modeled by trigonometric functions?

*How can you verify a trigonometric identity?