12th grade trigonometry

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

1sin( ) csc( )

u u

 1cos( ) sec( )

u u

 1tan( ) cot( )

u u



1csc( ) sin( )

u u

 1sec( ) cos( )

u u

 1cot( ) tan( )

u u



Quotient Identities

sin( )tan( ) cos( )

uu u

 cos( )cot( ) sin( )

uu u



Pythagorean Identities

2 2sin ( ) cos ( ) 1u u  2 21 tan ( ) sec ( )u u  2 21 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 2 2cot ( ) 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 4sin ( ) 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.

2sec ( )tan( ) tan( )

xx x



Ex. 7 (Rewriting a Trigonometric Expression) < F.TF.C* >

Rewrite the expression 2sin ( )

1 cos( ) y

y so that it is not in fractional

form.

Trigonometry/Precalculus Name

§5.1 Using Fundamental Identities HOMEWORK Date Pd

In Exercises 1-2, use the fundamental identities to simplify the expression. Verify with a graphing calculator. < F.TF.C* > 1. cot( )sin( )x x

2. sec( ) csc( )

 

In Exercises 3-4, verify the identity algebraically. Verify with a graphing calculator. < F.TF.C* > 3. 2 2sin( )csc( ) sin ( ) cos ( )    

4. 2sin ( )1 cos( )

1 cos( )   

   

In Exercises 5-6, factor the expression and use the fundamental identities to simplify. Verify with a graphing calculator. < F.TF.C* > 5. 2 2 2sec ( ) tan ( ) sec ( )x x x

6. 2cos ( ) 4

cos( ) 2 x x

 

In Exercises 7-8, perform the indicated operation and use the fundamental identities to simplify. < F.TF.C* > 7.  2sin( ) cos( )x x

8. 1 1 1 cos( ) 1 cos( )x x

  

In Exercises 9-10, rewrite the expression so that it is not in fractional form. < F.TF.C* >

9. 5 tan( ) sec( )x x

10. 2tan ( )

csc( ) 1 x

x 