Unit 5: Trigonometric Equations and Identities
Goal: The student will demonstrate the ability to solve trigonometric equations, prove and apply trigonometric identities.

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Objectives – The student will be able to:


a. Apply strategies to prove identities, including Pythagorean, and even and odd identities.

Mathematical Background/Clarifying Examples:
Students should be able to use basic identities and algebraic techniques to establish more complex identities.

Proving_Trig_Identities.PNG

Resources:

Website Links:
1. Trigonometric Identities: This website provides the most commonly used trigonometric identities.
http://www.purplemath.com/modules/idents.htm

2. Verifying Identities Practice: This website provides a few identities and their corresponding proofs.
http://www.analyzemath.com/trigonometry_worksheets.html



b. Verify trigonometric identities graphically.

Mathematical Background/Clarifying Examples:
The focus should be on the multiple representations of trigonometric expressions. Students should be able to demonstrate, using a table of values or a technologically-produced graph, that two trigonometric expressions are equivalent.

Resources:

Website Links:
1. Using the Graphing Calculator to Verify Identities: This website provides instructions for graphing trigonometric identities.
http://mathbits.com/mathbits/tisection/trig/trigidentity.htm


c. Use the addition and subtraction identities for sine, cosine, and tangent functions.

Mathematical Background/Clarifying Examples:
Sum and Difference Formulas for sine, cosine, and tangent should be used to find exact values as well as to establish trigonometric identities.

Finding the exact value using Sum Formula:
Finding_Exact_Value.PNG
Proving Identities using Sum and Difference Formulas:

Sum_and_Diff_Proofs.PNG

Resources:

Website Links:
1. Sum and Difference Formulas: This website provides a list of the Sum and Difference Formulas.
Sum and Difference Formulas


d. Use the double-angle and half-angle identities.

Mathematical Background/Clarifying Examples:
Half-angle Identities and Double-Angle Identities should be used to find exact values. Double-Angle Identities, but not Half-Angle Identities, should also be used to verify other identities. Half-Angle Identities may be used in proofs as a possible extension.

Finding exact values using Half-Angle Identities.
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Finding exact values using Double-Angle Identities.
Proving_Id_with_Double_Angle_ID.PNG

Verifying Identities using Double-Angle Identities:

Proving_Id_with_Double_Angle_ID_2.png

Resources:

Website Links:
1. Double-Angle and Half-Angle Formulas: This website provides explanations of these identities and examples of their uses, with detailed solutions.
http://math.ucsd.edu/%7Ewgarner/math4c/textbook/chapter6/doublehalfangles.htm


e. Use identities to solve trigonometric equations.

Mathematical Background/Clarifying Examples:
Students should be able to use previously learned identities in order to solve trigonometric equations. It is essential that students are comfortable with the identities so that they are able to determine reasonable identities to use for given equations.
Solving_Trig_Equations_Using_Identities.PNG

Resources:
Files:
1. Solving Equations Using Identities: This file provides practice for solving trigonometric equations using given identities.




f. Solve trigonometric equations graphically and algebraically.

Mathematical Background/Clarifying Examples:
Students should be able to manipulate trigonometric equations using various algebraic methods such as factoring in order to arrive at solutions. It is important to use graphing and tables of values as means to verify solutions to trigonometric equations.
Solving_by_Linear_Methods.PNG

Solving_by_Factoring.PNG
Solving_by_Quadratic.PNG

Solving_by_Squaring.PNG
Solving a Trigonometric Equation Graphically

Resources:

Files:
1. Solving Trig Equations: This file provides explanations of how to solve trigonometric equations using graphical and algebraic methods as well as practice with these methods.

2. Real world example: This provides an example of how trigonometric equations can be applied to real world scenarios.
Screen_shot_2011-07-01_at_11.36.10_AM.png