Transcript 5 1 Fundamental Identities 01

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P. 379: 15-20
P. 379: 21-26
P. 379: 27-44 S
P. 380: 61-64 S
Homework Supplement
You may remember
these identities
fondly from
previous lessons.
Now we’ll be using
them to simplify
and rewrite trig
expressions.
Practice doing this
will make solving
trig equations
easier.
Objective:
1. To rewrite and
simplify trig
expressions using
trig identities
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•
•
•
•
Assignment:
P. 379: 15-20
P. 379: 21-26
P. 379: 27-44 S
P. 380: 61-64 S
Homework
Supplement
Remember that the cofunctions of
complementary angles are equal.
Cofunctions of complementary angles are equal.
sin   cos  90   
cos   sin  90   
tan   cot  90   
cot   tan  90   
sec  csc  90   
csc  sec  90   
*An identity is an equation that is true for all values in the domain.
Cofunctions of complementary angles are equal.


sin   cos    
2



cos   sin    
2



tan   cot    
2



cot   tan    
2



sec   csc    
2



csc   sec    
2

*An identity is an equation that is true for all values in the domain.
Half of the trig functions are reciprocals of the
other half.
1
sin  
csc 
1
cos  
sec 
1
tan  
cot 
1
csc  
sin 
1
sec  
cos 
1
cot  
tan 
Notice that it is not the cofuctions that are
reciprocals: C w/S or T.
Tangent and cotangent are quotients of sine and
cosine.
sin 
tan  
cos 
cos 
cot  
sin 
These are easy to verify if you use the unit circle
definitions of sine, cosine, and tangent:
sin   y
cos  x
y
sin 
tan  

x
cos 
The following identities are based on the
Pythagorean Theorem.
sin 2   cos 2   1
1  tan 2   sec 2 
1  cot 2   csc2 
Note that sin2 θ = (sin θ)2
Again, verifying these identities is pretty easy
using the unit circle definitions:
sin   y
cos  x
By the Pythagorean Theorem:
x2  y 2  1
cos 2   sin 2   1
For the other P-Thag Identities, just divide
everything by sin2 θ or cos2 θ and simplify.
sin 2   cos 2   1
sin 2   cos 2   1
sin 2  cos 2 
1


2
2
sin  sin  sin 2 
sin 2  cos 2 
1


2
2
cos  cos  cos 2 
1  cot 2   csc2 
tan 2   1  sec 2 
In the course of simplifying some expressions,
you may run across some alternate versions:
sin 2   cos 2   1
1  tan 2   sec 2 
1  cot 2   csc2 
sin 2   1  cos 2 
tan 2   sec 2   1
cot 2   csc2   1
sin    1  cos2 
tan    sec2   1
cot    csc2   1
cos 2   1  sin 2 
sec   1  tan 2 
csc   1  cot 2 
cos    1  sin 2 
sec 2   tan 2   1
csc2   cot 2   1
• Cosine and Secant are even:
cos(t )  cos t
sec(t )  sec t
• Sine, Cosecant, Tangent, and Cotangent are
odd:
sin(t )   sin t
csc(t )   csc t
tan(t )   tan t
cot(t )   cot t
Find each of the following:
cos(30)
sin(30)
cos(30)
sin(30)
When simplifying and rewriting trig expressions,
here are some things to try:
1. Substitute part of the expression with an
identity
2. Factor or FOIL
3. Write as a single trig function
4. Rewrite as sine and cosine only
5. Combine fractions
Simplify:
cos 2 x csc x  csc x
Simplify:
sin x cos 2 x  sin x
Simplify:
1  sin x 1  sin x 
Simplify:
1  cos x 1  cos x 
Simplify:
cos  tan   sec 
Simplify:
sin   cot   csc 
When you add or subtract two fractions, you
need to get a common denominator, of
course:
1 1 1
x

y

z
Simplify:
1
1

1  sin x 1  sin x
Simplify:
sin x
cos x

1  cos x sin x
Simplify:
csc t  cos t cot t
Simplify:
sin t  cot t cos t
Complex fractions are fractions within fractions.
Here’s a few ways to deal with them.
1. Turn Division into Multiplication
k
1
k
k1
 k k
 k2
k
1
2. Multiply by the LCD on Top/Bottom
k
k

1
k
k
k2

1
 k2
Complex fractions are fractions within fractions.
Here’s a few ways to deal with them.
This next method, I’ll call TB-BT, or
Top/Bottom-Bottom/Top.
1. Numerator:
Number stays on Top
Number goes to the Bottom
2.
Denominator:
Number stays on Bottom
Number goes to the Top
3. TB-BT
k
k
 1
1
1
k
k
k2

1
 k2
Simplify:
tan 
cot 
Simplify:
sec 
csc 
Simplify:
cos 2 t
1
cot 2 t
Objective:
1. To rewrite and
simplify trig
expressions using
trig identities
•
•
•
•
•
Assignment:
P. 379: 15-20
P. 379: 21-26
P. 379: 27-44 S
P. 380: 61-64 S
Homework
Supplement