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 557You 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. 2. Q:  Verify the identity.

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. 2. Q:  Find the exact value of the expression. 3. Q:  Find the exact value of the expression.

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 . 2. Q:   Write as the tangent of a double angle.  Then find the exact value of the expression. 3. Q:   If  and lies in quadrant II, find .

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. 2. Q:  Express the difference as a product.

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.

1)  Isolate the function on one side of the equation.

2)  Solve for the variable.

3.  It is important to know the period of the trig. function so that you do not forget any of the equation's solutionsTo every solution that you find , you must add . You will keep doing this until you exceed any restriction given  in the problem for the domain.

4.  When solving a multiple angle problem, remember that the last step is to solve for the variable.  For example, if the argument of the function is , you would need to divide both sides of the equation by 2.

5.  Some trigonometric equations may be in a quadratic form.  This means that some may be solved by factoring, but others will require the use of the quadratic formula.  A review of the quadratic formula can be found beginning on page 121.

6.   When a trigonometric equation contains more than one function on the same side, it may be necessary to use an identity before you can solve the equation.

7.   Remember!  If a trigonometric equation contains the same trig. function on both sides, you cannot divide by that function on both sides and cancel it out.

 Additional Exercises: 1. Q:  Find all the solutions to the equation. 2. Q:  Solve the equation on the given interval.

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