Transcript Lesson 5.1
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
Quadrant I
Sin :
Sin :
Cos :
Tan :
Cos :
Tan :
Quadrant III
Quadrant IV
Sin :
Sin :
Cos :
Tan :
Cos :
Tan :
5.1 Using Fundamental Identities
Using Identities to Evaluate a Function
Given:
3
Sec
2
Tan 0
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
5.1 Using Fundamental Identities
Given: Sec 3
2
Tan 0
*Use the information given to decide which quadrant
Which quadrant are we in?1______
Find: The value of all six trigonometric functions
1
Cos
Sec
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 Identities
Pythagorean Identity2
Sin Cos 1
2
2
5.1 Using Fundamental Identities
Now we know3 :
5
Sin
3
2
Cos
3
sin
Tan
cos
1
Csc
sin
1
Sec
cos
1
Cot
t an
5.1 Using Fundamental Identities
Now try #11 pg.3454
5.1 Using Fundamental Identities
Simplifying a Trigonometric Expression5
sin x cos x sin x
2
Factor out a sin
x
Use the distributive property
Use the Pythagorean Identity
Multiply
5.1 Using Fundamental Identities
Verifying a Trigonometric Identity7
sin
cos
csc
1 cos sin
(sin )(sin ) (cos )(1 cos )
csc
(1 cos )(sin )
5.1 Using Fundamental Identities
Now 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
style for y 2 . )
y1
y 2 . (Select line style for y1 and path
5.1 Using Fundamental Identities
Factoring Trigonometric Expressions9
a.)
sec 1
2
b.)
4 tan tan 3
2
5.1 Using Fundamental Identities
Now 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.
cost
Simplify11: sin t cott cost sin t
cost
sin t
5.1 Using Fundamental Identities
Now try #67 pg.34612
5.1 Using Fundamental Identities
Enriched Pre-Calculus
Rewriting a Trigonometric Equation
Rewrite13
1
1 sin x
so that it is not in fractional form.
1
1
1 sin x
1 sin x 1 sin x 1 sin x
5.1 Using Fundamental Identities
Now try #69 pg.34614
5.1 Using Fundamental Identities
Enriched Pre-Calculus
Trigonometric Substitutions15
Use the substitution
x 2 tan ,0
as a trigonometric function of
.
4 x 4 (2 tan )
2
2
2
to write 4 x2
5.1 Using Fundamental Identities
Now try #81 pg.34716
THE
END