### Course 4 Unit 4 - Trigonometric Functions and Equations ©2010

In Core-Plus Mathematics Course 2 Unit 6, Trigonometric Methods, students learned right triangle trigonometry, the Law of Sines, and the Law of Cosines. The sine and cosine function work was mainly with angles measured in degrees in the first quadrant although some tasks considered angles in other quadrants. In Course 3 Unit 6, Circles and Circular Functions, the work with circular functions, sine, cosine, and tangent functions was extended to all quadrants and radian measure was introduced. In Unit 8, Inverse Functions, students developed an understanding of the inverse functions for the sine, cosine, and tangent and then used inverse trigonometric functions to solve application problems. In Course 4 Unit 1, students reviewed the sine and cosine functions in the context of families of functions. (See the CPMP Courses 1-4 descriptions.)

Unit Overview
In this unit, students review and extend their understanding of trigonometric functions. The development emphasizes symbolic reasoning with equivalent expressions in the contexts of identities and equations involving trigonometric functions. Students manipulate the symbolic representations to obtain equivalent representations that may be "simpler" or more easily interpreted, graphed, or used. In addition, complex numbers are revisited and connected to trigonometric functions, identities, and polar coordinates. A major goal of this unit is to help students become proficient in symbolic reasoning, rewriting trigonometric expressions into equivalent forms, and using trigonometric relations to understand the geometry of complex numbers.

 Objectives of the Unit Know and be able to use the definitions of the six trigonometric functions of an angle in standard position Derive and use the fundamental trigonometric identities Prove trigonometric identities Solve trigonometric equations Represent complex numbers geometrically Interpret multiplication and division of complex numbers geometrically Use De Moivre's Theorem to find powers and roots of complex numbers

Sample Overview
The sample material below is from Lesson 3, "The Geometry of Complex Numbers." This second investigation draws on the process of multiplying complex numbers developed in Investigation 1 to develop the connection between multiplication of complex number expressed in trigonometric form and transformations of the coordinate plane (De Moirvre's Theorem).
This content is recommended in the Common Core State Standards (CCSS) for Mathematics for students planning to concentrate in mathematics and science in postsecondary programs. See: N-CN, #5 of the CCSS document.

Instructional Design
Throughout the curriculum, interesting problem contexts serve as the foundation for instruction. As lessons unfold around these problem situations, classroom instruction tends to follow a four-phase cycle of classroom activities—Launch, Explore, Share and Summarize, and Apply. This instructional model is elaborated under Instructional Design.