Chapter 5.1

5.1 Using Fundamental Identities

In this chapter, you will learn how to use the fundamental identities to do the following:

Evaluate trigonometric functions

Simplify trigonometric expressions

Develop additional trigonometric identities

Solve trigonometric equations

5.1 Using Fundamental Identities

Using Identities to evaluate a function

Why should you learn this?

•In calculus you will use trigonometric identities to simplify an expression.

•Ex: Homework p.347 #97 is an example

5.1 Using Fundamental Identities

Quadrant II

:

:

:

Tan

Cos

Sin

Quadrant IVQuadrant III

Quadrant I

:

:

:

Tan

Cos

Sin

:

:

:

Tan

Cos

Sin

:

:

:

Tan

Cos

Sin

5.1 Using Fundamental IdentitiesUsing Identities to Evaluate a Function

Given:

Find: The value of all six trigonometric functions

*Remember:

Once you know the value of Sine and Cosine you can find the values of all six trigonometric functions

02

3

Tan

Sec

5.1 Using Fundamental IdentitiesGiven:

*Use the information given to decide which quadrant

Which quadrant are we in?1______

Find: The value of all six trigonometric functions

02

3

Tan

Sec

Sec

Cos1

Looking at page 340 in your book:

Can you see which identity we should use to find the value of sine?

5.1 Using Fundamental IdentitiesPythagorean Identity2

122 CosSin

5.1 Using Fundamental IdentitiesNow we know3 :

cos

sinTan

3

23

5

Cos

Sin

tan

1

cos

1

sin

1

Cot

Sec

Csc

5.1 Using Fundamental IdentitiesNow try #11 pg.3454

5.1 Using Fundamental IdentitiesSimplifying a Trigonometric Expression5

xxx sincossin 2 Factor out a

Use the distributive property

Use the Pythagorean Identity

Multiply

xsin

5.1 Using Fundamental IdentitiesVerifying a Trigonometric Identity7

cscsin

cos

cos1

sin

csc

))(sincos1(

)cos1)((cos))(sin(sin

5.1 Using Fundamental IdentitiesNow try #45 pg.3468

To check graphically, use radian mode graph one side of the equation as

and the other side of the equation as . (Select line style for and path style for . )

1y

2y1y2y

5.1 Using Fundamental IdentitiesFactoring Trigonometric Expressions9

1sec2 3tantan4 2 a.) b.)

5.1 Using Fundamental IdentitiesNow try #51 pg.34610

5.1 Using Fundamental Identities*NOTE: On occasion, factoring or simplifying can best be done by rewriting the expression in terms of just one trigonometric function or in terms of sine or cosine alone.

Simplify11: tt

ttttt cos

sin

cossincoscotsin

5.1 Using Fundamental IdentitiesNow try #67 pg.34612

5.1 Using Fundamental IdentitiesEnriched Pre-Calculus

Rewriting a Trigonometric Equation

Rewrite13 so that it is not in fractional form.xsin1

1

x

x

xx sin1

sin1

sin1

1

sin1

1

5.1 Using Fundamental IdentitiesNow try #69 pg.34614

5.1 Using Fundamental IdentitiesEnriched Pre-Calculus

Trigonometric Substitutions15

Use the substitution to write

as a trigonometric function of .2

0,tan2 x 24 x

22 )tan2(44 x

5.1 Using Fundamental IdentitiesNow try #81 pg.34716

THE END