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