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Chapter 6: Analytic Trigonometry Study Guide

Section 6.1
Hints:

1.   You must memorize the Fundamental Identities that are given in the box on page 557.  You may also need to algebraically manipulate these identities.

            Ex.: 

2.  To verify an identity, you show that one side of the identity can be simplified so that it is identical to the other side.

            Ex.: 

In the example above, the left side of the identity is worked on until it looks like the right side.

3.  Remember!  Since you are verifying that the identity is true, you cannot treat it like an equation.  You cannot add a term to or subtract a term from both sides of the identity.  You cannot multiply or divide both sides of an identity by an expression.

            Ex.:  

In the example above, it is incorrect to subtract from both sides of the identity.

4.   It is permissible to perform algebraic simplification one side of the identity.  For example, you may need to factor one side of the identity or combine two terms with a common denominator.

            Ex.:   

5.   It is all right to work on both sides of an identity at the same time, in order to turn both sides into the same expression.

6.  There is a set of guidelines for verifying trigonometric identities on page 565.  These will be very helpful.

Additional Exercises:
1.	Q:  Verify the identity.       Click Here for Answer
2.	Q:  Verify the identity.       Click Here for Answer

Section 6.2
Hints:

1.  The Sum Identities are given below.

  
2.   The Difference Identities are given below.

3.  These identities can be used to find the exact values of trigonometric functions when the angle can be written as the sum or difference of  two special angles.

        Ex.:  

 

    

Additional Exercises:
1.	Q:  Find the exact value of the expression.       Click Here for Answer
2.	Q:  Find the exact value of the expression.       Click Here for Answer
3.	Q:  Find the exact value of the expression.       Click Here for Answer

 

Section 6.3
Hints:

1.   There are three Double Angle identities.

 

 

 

2.  The double-angle formula for  can then be written in two additional forms.

 

3.  The double-angle formulas are used to derive the power-reducing formulas.

 

4.  Useful forms of the power-reducing formulas can be found by replacing with .  These are called the half-angle formulas.

 

The + or - in each formula is determined by the quadrant in which lies.

 

5.  On page 585, there is a helpful summary of the all the identities found in this section.

Additional Exercises:
1.	Q:  If and lies in quadrant IV, find .        Click Here for Answer
2.	Q:   Write as the tangent of a double angle.  Then find the exact value of the expression.        Click Here for Answer
3.	Q:   If  and lies in quadrant II, find .       Click Here for Answer

Section 6.4
Hints:

1.  The products of sines and/ or cosines can be written as sums or differences.  These identities are called product-to-sum formulas.

 

     

 

2.   The sum or difference of sines and/or cosines can be written as products.  These identities are called sum-to-product formulas.

 

3.  Some identities contain a fraction on one side with sums and differences of sines and/or cosines.  It is often helpful in verifying these identities to use one on the sum-to-product formulas.

 

 

Additional Exercises:
1.	Q:  Express the product as a sum or difference.       Click Here for Answer
2.	Q:  Express the difference as a product.       Click Here for Answer

Section 6.5
Hints:

1.  A trigonometric equation is an equation that contains a trigonometric expression with a variable.  Because of the periodic natural of trig. functions, trigonometric equations can have infinitely many solutions.

2.  To solve an equation containing a single trig. function, follow the steps given below.