Unit 4: Inverse Circular and Trigonometric Functions Goal: The student will demonstrate the ability to investigate and apply inverse circular and inverse trigonometric functions in order to prove basic identities.

Objectives – The student will be able to: Note: The circular functions are just like the trigonometric functions except that the independent variable is an arc of a unit circle instead of an angle. a. Define the domain and range of the inverse circular functions.

Mathematical Background:
Build on students previous knowledge of inverse functions to develop the inverse trigonometric functions. Discuss possible interval restrictions on the domain that would make each inverse a function. Finally, discuss the reasonable domain restrictions of y=arcsin x, y=arccos x, and y=arctan x so that it is a one to one function.

Clarifying Examples:

Resources: Tutorial, Interactive Practice, and Examples: This site explains the inverse functions for sine, cosine, and tangent, gives 4-6 interactive practice problems for each. Additional examples are provided. http://www.themathpage.com/atrig/inverseTrig.htm

c. Define the domain and range of the inverse trigonometric functions and sketch the graph.

Mathematical Background:
Students will need to identify the restrictions on the domain and range for the inverse trigonometric functions and sketch the curve on that interval.

Mathematical Background:
Students will use knowledge of the six trigonometric ratios to evaluate expressions. Real-world application problems should be incorporated into this objective (see objective e).

e. Use inverse functions to model and solve real-world problems.

Mathematical Background:
Students need to use knowledge from previous sections to solve for angles in real-world applications.

Clarifying Examples:
A dump truck's bed is 10 ft long. Find the angle it needs to be lifted in order for it to reach a height of 7 ft from it's original position.

Unit 4: Inverse Circular and Trigonometric FunctionsGoal: The student will demonstrate the ability to investigate and apply inverse circular and inverse trigonometric functions in order to prove basic identities.Objectives – The student will be able to:

Note: The circular functions are just like the trigonometric functions except that the independent variable is an arc of a unit circle instead of an angle.

a. Define the domain and range of the inverse circular functions.Mathematical Background:Build on students previous knowledge of inverse functions to develop the inverse trigonometric functions. Discuss possible interval restrictions on the domain that would make each inverse a function. Finally, discuss the reasonable domain restrictions of y=arcsin

x, y=arccosx, and y=arctanxso that it is a one to one function.Clarifying Examples:Resources:Tutorial, Interactive Practice, and Examples:This site explains the inverse functions for sine, cosine, and tangent, gives 4-6 interactive practice problems for each. Additional examples are provided.http://www.themathpage.com/atrig/inverseTrig.htm

Applet: This site shows three graphs side by side: the trig curve, it's inverse, and then the composition of the functions for sine, cosine and tangent.http://orion.math.iastate.edu/trig/sp/xcurrent/applets/inversetrig.html

Tutorial, Instructional Tools, and Examples:This site gives tutorial and memory tools for all six inverse trig functions, with great visuals for learners. The site also includes examples which can be used in various places for other objectives in this unit.http://www.sinclair.edu/centers/mathlab/pub/findyourcourse/worksheets/116,117/InverseTrigFunctions.pdf

b. Evaluate the inverse circular functions.Mathematical Background:Students should build upon their knowledge of the unit circle and angles to simplify given expressions.

Clarifying Examples:Resources:Inverse Functions Practice:This resource provides practice problems for using inverse circular functions.c. Define the domain and range of the inverse trigonometric functions and sketch the graph.Mathematical Background:Students will need to identify the restrictions on the domain and range for the inverse trigonometric functions and sketch the curve on that interval.

Tutorial: This site discusses the inverses of all six trigonometric functions and gives numerous graphs in the explanation.http://www.intmath.com/analytic-trigonometry/7-inverse-trigo-functions.php

Graphing calculator exploration:This site provides directions and questions to ask the class during the activity to promote learning and critical thinking.http://www.regentsprep.org/Regents/math/algtrig/ATT8/inversetrigResource.htm

Graphs shown with inverses:This site shows trigonometric graphs with their inverses, as well as the domain sets for each function.http://hotmath.com/hotmath_help/topics/inverse-trigonometric-functions.html

d. Evaluate the inverse trigonometric functions.Mathematical Background:Students will use knowledge of the six trigonometric ratios to evaluate expressions. Real-world application problems should be incorporated into this objective (see objective e).

Clarifying Examples:Resources:Interactive practice:This site provides practice for objectives a-d.http://www.regentsprep.org/Regents/math/algtrig/ATT8/Practiceinv.htm

Online student practice:This site gives students online flashcards to practice using inverse trig functions.http://quizlet.com/4085516/evaluate-inverse-trig-functions-flash-cards/

e. Use inverse functions to model and solve real-world problems.Mathematical Background:Students need to use knowledge from previous sections to solve for angles in real-world applications.

Clarifying Examples:A dump truck's bed is 10 ft long. Find the angle it needs to be lifted in order for it to reach a height of 7 ft from it's original position.

Example: This site gives an example using inverse circular functions and includes the graphs.http://www.scs.sk.ca/hch/harbidge/C30/Unit%203/lesson_6/problems.htm

Resources:Website:Solving for Angles:This website has an example of using inverse trig functions to find an angle in degrees.http://mathcentral.uregina.ca/QQ/database/QQ.09.03/jason1.html