In chapter 4, you studied the basic definitions, properties, graphs, and applications of the individual trigonometric functions. In this chapter, you will learn how to use the fundamental identities to do the following. 1. Evaluate trigonometric functions. 2. Simplify trigonometric expressions.

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Trigonometry/Precalculus Name
§5.1 Using Fundamental Identities Date Pd
Introduction
In chapter 4, you studied the basic definitions, properties, graphs,
and applications of the individual trigonometric functions. In this
chapter, you will learn how to use the fundamental identities to do
the following.
1. Evaluate trigonometric functions.
2. Simplify trigonometric expressions.
3. Develop additional trigonometric identities.
4. Solve trigonometric equations.
Fundamental Trigonometric Identities
Reciprocal Identities
1 sin( ) csc( )
u
u  1 cos( ) sec( )
u
u  1 tan( ) cot( )
u
u 
1 csc( ) sin( )
u
u  1 sec( )
cos( )
u
u  1 cot( ) tan( )
u
u 
Quotient Identities
sin( ) tan( )
cos( )
u
u
u  cos( ) cot( ) sin( )
u
u
u 
Pythagorean Identities
2 2 sin ( ) cos ( ) 1 u u   2 2 1 tan ( ) sec ( )   u u
2 2 1 cot ( ) csc ( )   u u
Ex. 1 (Simplifying a Trigonometric Expression) < F.TF.C* >
Use the fundamental identities to simplify the expression
sin( ) csc( ) sin( ) .      
Example 2 (Verifying a Trigonometric Identity) < F.TF.C* >
Verify the identity algebraically. Verify with a graphing calculator.
sin( )   cos( )   cot( )  csc( )
Example 3 (Verifying a Trigonometric Identity) < F.TF.C* >
Verify the identity algebraically. Verify with a graphing calculator.
csc( )  tan( )    sec( )
Ex. 4 (Factoring a Trigonometric Expression) < F.TF.C* >
Factor the expression 2 22 cot ( ) cot ( )cos ( ) x  x x and use the
fundamental identities to simplify. Verify with a graphing calculator.
Ex. 5 (Factoring a Trigonometric Expression) < F.TF.C* >
Factor the expression 4 4 sin ( ) cos ( ) x  x and use the fundamental
identities to simplify. Verify with a graphing calculator.
Ex. 6 (Simplifying a Trigonometric Expression) < F.TF.C* >
Perform the indicated operation and use the fundamental identities
to simplify.
2 sec ( ) tan( ) tan( )
x
x
x 
Ex. 7 (Rewriting a Trigonometric Expression) < F.TF.C