Unit 2
Trigonometric Functions

 

Algebra II

Unit Topic and Length:

 

20 days

 

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. (Mathematics Appendix A, p.36-40, with adjustments)

 

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

·        Model periodic phenomena with trigonometric function.

·        Prove and apply trigonometric identities.

 

Standards:

 

F-TF.A.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

 

  • Standards Clarification:  Know that if the length of an arc subtended by an angle is the same length as the radius of the circle, then the measure of the angle is 1 radian.  Know that the graph of the function, f is the graph of the equation y = f(x).

 

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.

 

  • Standards Clarification:  Students may believe that there is no need for radians if one already knows how to use degrees. Students need to be shown a rationale for how radians are unique in terms of finding function values in trigonometry since the radius of the unit circle is 1.  Students may also believe that all angles having the same reference values have identical sine, cosine and tangent values. They will need to explore in which quadrants these values are positive and negative. Also extend trigonometric functions to their reciprocal functions.

 

F-TF.A.3(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for /3,  /4 and  /6, and use the unit circle to express the values of sine, cosine, and tangent for –x, +x, and 2 –x in terms of their values for x, where x is any real number.

 

  • Standards Clarification:  For this “+” standard, use 30°-60°-90° and 45°-45°-90° triangles to determine the values of sine, cosine, and tangent for values of /3,  /4 and  /6.

 

F-TF.B.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

 

  • Standards Clarification:  Use sine and cosine to model periodic phenomena such as the ocean’s tide or the rotation of a Ferris wheel.  Given the amplitude; frequency; and midline in situations or graphs, determine a trigonometric function used to model the situation.  Allow students to explore real-world examples of periodic functions. Examples include average high (or low) temperatures throughout the year, the height of ocean tides as they advance and recede, and the fractional part of the moon that one can see on each day of the month. Graphing some real-world examples can allow students to express the amplitude, frequency, and midline of each. Help students to understand what the value of the sine (cosine, or tangent) means (e.g., that the number represents the ratio of two sides of a right triangle having that angle measure).  Using graphing calculators or computer software, as well as graphing simple examples by hand, have students graph a variety of trigonometric functions in which the amplitude, frequency, and/or midline is changed. Students should be able to generalize about parameter changes, such as what happens to the graph of y = cos(x) when the equation is changed to y = 3cos(x) + 5.  Students may believe that all trigonometric functions have a range of 1 to -1. Students need to see examples of how coefficients can change the range and the look of the graph.

 

F-TF.C.8 Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 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.

 

  • Standards Clarification:  In the unit circle, the cosine is the x-value, while the sine is the y-value. Since the hypotenuse is always 1, the Pythagorean relationship sin2 (θ) + cos2 (θ) = 1 is always true. Students can make a connection between the Pythagorean Theorem in geometry and the study of trigonometry by proving this relationship. In turn, the relationship can be used to find the cosine when the sine is known, and vice-versa. Provide a context in which students can practice and apply skills of simplifying radicals.  Students may believe that there is no connection between the Pythagorean Theorem and the study of trigonometry.  Students may also believe that there is no relationship between the sine and cosine values for a particular angle. The fact that the sum of the squares of these values always equals 1 provides a unique way to view trigonometry through the lens of geometry.

 

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

 

  • Standards Clarification:  For Algebra II, this “+” standard could be limited to acute angles.

 

Focus Standards of 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.

 

Instructional Outcomes:

Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:

 

F-TF.A.1

·        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

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

 

F-TF.A.3(+)

·        I can find the exact value of trigonometric functions using the unit circle.

·        I can verify and identify trigonometric identities.

·        I can use special triangles to determine geometrically the values of sine, cosine, tangent for /3, /4 and /6.

 

F-TF.B.5

·        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.

 

F-TF.C.8

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

·        I can prove the Pythagorean identity sin2(Ө) + cos2(Ө) =1.

·         I can use the Pythagorean identity, sin2(Ө) + cos2(Ө) =1, to find sin (Ө), cos (Ө), or tan (Ө), given sin(Ө), cos(Ө), or tan(Ө), and the quadrant of the angle.

 

F-TF.C.9(+)

·        I can verify and identify trigonometric identities.

·        I can prove the addition and subtractions formulas for sine, cosine, and tangent.

·        I can use the addition and subtractions formulas for sine, cosine, and tangent to solve problems.

 

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 sin2(θ) + cos2(θ) = 1 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?