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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
