Course 2 Unit 7 - Trigonometric Methods

Trigonometry or "measure of triangles" is an important and useful area of mathematics that naturally connects concepts and methods of geometry and algebra. Trigonometric Methods builds on the Course 1 geometry unit, Patterns in Shape, and the Course 2 unit, Coordinate Methods. In Course 1, students explored the rigidity of triangles and minimal conditions that are sufficient to completely determine a triangle's size and shape. Once a triangle is completely determined, a next logical question is "How can the measures of the triangle's unknown, but rigidly determined, sides and angles be calculated?". Trigonometry provides those methods, namely, trigonometric ratios for right triangles and the Law of Sines and Law of Cosines for any triangle.

Unit Overview
In this unit, students develop the ability to use right triangle trigonometry to solve triangulation and indirect measurement problems. The also begin to develop an understanding of trigonometric functions.

Objectives of the Unit
  • Explore the sine, cosine, and tangent functions defined in terms of a point on the terminal side of an angle in standard position in a coordinate plane
  • Explore properties and applications of the sine, cosine, and tangent ratios of acute angles in right triangles
  • Determine measures of sides and angles for nonright triangles using the Law of Sines and Law of Cosines
  • Use the Law of Sines and Law of Cosines to solve a variety of applied problems that involve triangulation
  • Describe the conditions under which two, one, or no triangles are determined given the lengths of two sides and the measure of an angle not included between the two sides

Sample Overview
The sample student material below is from Lesson 2, "Using Trigonometry in Any Triangle." Students prove and use the Law of Sines in this investigation. Embedded in this work is solving proportions.

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.

View the Unit Table of Contents and Sample Lesson Material
You will need the free Adobe Acrobat Reader software to view and print the sample material.

How the Geometry and Trigonometry Strand Continues
In Course 3 Units 1 and 3, students extend their ability to reason formally in geometric settings. Deductive reasoning is used to prove theorems concerning parallel lines and transversals, angle sums of polygons, similar and congruent triangles and their application to special quadrilaterals, and necessary and sufficient conditions for parallelograms. Circular functions (sine and cosine) are used to model periodic change in Unit 6, Circles and Circular Functions.
     In Course 4: Preparation for Calculus, geometry and algebra become increasingly intertwined. Students develop understanding of two-dimensional vectors and their application and the use of parametric equations in modeling linear, circular, and other nonlinear motion. In addition, students intending to pursue programs in the mathematical, physical, and biological sciences, or engineering extend their ability to visualize and represent three-dimensional surfaces using contours, cross sections, and reliefs; and to visualize and sketch surfaces and conic sections defined by algebraic equations. They also extend their understanding of, and ability to reason with, trigonometric functions to prove or disprove trigonometric identities and to solve trigonometric equations. They also geometrically represent complex numbers and apply complex number operations to find powers and roots of complex numbers expressed in trigonometric form. (See the CPMP Courses 1-4 descriptions.)

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