Download Sec 8.4 Simplify and Prove trig

While you wait:
For a-d: use a calculator to evaluate:
a) sin 50𝑜 , cos 40𝑜
b) sin 25𝑜 , 𝑐𝑜𝑠65𝑜
c) 𝑐𝑜𝑠11𝑜 , sin 79𝑜
d) sin 83𝑜 , cos 7𝑜
Fill in the blank.
a) 𝑠𝑖𝑛30° = cos ___°
b) 𝑐𝑜𝑠57° = sin ___°
Trigonometric
Identities and
Equations
Section 8.4
Cofuntion Relationships
UC revisited
y
(x,y)
1
y
x
x
Pythagorean Theorem: 𝑥 2 + 𝑦 2 = 1
UC revisited
y
(cos , sin  )
1
θ
sin 
x
cos
Pythagorean Theorem:𝑐𝑜𝑠 2 𝜃 + 𝑠𝑖𝑛2 𝜃 = 1
The trig relationships:
𝑐𝑜𝑠 2 𝜃 + 𝑠𝑖𝑛2 𝜃 = 1
𝑐𝑜𝑠 2 𝜃 = 1 − 𝑠𝑖𝑛2 𝜃
𝑠𝑖𝑛2 𝜃 = 1 − 𝑐𝑜𝑠 2 𝜃
• An identity is an equation that is true for all
values of the variables.
• Difference between identity and equation:
• An identity is true for any value of the variable, but
an equation is not. For example the equation 3x=12
is true only when x=4, so it is an equation, but not an
identity.
What are identities used for?
• They are used in simplifying or rearranging
algebraic expressions.
• By definition, the two sides of an identity are
interchangeable, so we can replace one with
the other at any time.
• In this section we will study identities with trig
functions.
The trigonometry identities
• There are dozens of identities in the field of
trigonometry.
• Many websites list the trig identities. Many
websites will also explain why identities are
true. i.e. prove the identities.
• For an example of such a site: click here
Trigonometric Identities
Quotient Identities
sin 
tan  
cos 
cos 
cot  
sin 
Reciprocal Identities
1
sin  
csc 
1
cos  
sec 
1
tan  
cot 
Pythagorean Identities
sin2 + cos2 = 1
tan2 + 1 = sec2
cot2 + 1 = csc2
sin2 = 1 - cos2
tan2 = sec2 - 1
cot2 = csc2 - 1
cos2 = 1 - sin2
5.4.3
Where did our pythagorean identities come from??
Do you remember the Unit Circle?
• What is the equation for the unit circle?
x2 + y2 = 1
• What does x = ? What does y = ?
(in terms of trig functions)
sin2θ + cos2θ = 1
Pythagorean
Identity!
Take the Pythagorean Identity and
discover a new one!
Hint: Try dividing everything by cos2θ
sin2θ + cos2θ = 1 .
cos2θ cos2θ cos2θ
tan2θ + 1 = sec2θ
Quotient
Identity
another
Pythagorean
Identity
Reciprocal
Identity
Take the Pythagorean Identity and
discover a new one!
Hint: Try dividing everything by sin2θ
sin2θ + cos2θ = 1 .
sin2θ sin2θ sin2θ
1 + cot2θ = csc2θ
Quotient
Identity
a third
Pythagorean
Identity
Reciprocal
Identity
Simplifying Trigonometric Expressions
Identities can be used to simplify trigonometric expressions.
Simplify.
a)
cos   sin  tan 
sin
 cos   sin 
cos 
sin2 
 cos  
cos 
cos   sin 

cos 
1

cos 
2
 sec 
2
cot 2 
1  sin2 
b)
cos 2 
2
sin


cos 2 
1
cos 2 
1


2
sin  cos2 

1
2
sin 
 csc 2 
5.4.5
• Practice Problems for Day 1:
refer to class handout.
While you wait
• Factor:
a) 𝑥 2 − 4
b) 𝑥 2 − 36
c) 𝑥 2 − 1
d) 1 − 𝑥 2
• Identify as True or False:
A. cos −𝜃 = −cos(𝜃)
B. sin −𝜃 = −𝑠𝑖𝑛(𝜃)
C. tan −𝜃 = −tan(𝜃)
Proving a Trigonometric Identity:
1. Transform the right side of the identity into the left
side,
2. Vice versa (Left side to Right )
We do not want to use properties from algebra
that involve both sides of the identity.
Guidelines for Proving Identities:
1. It is usually best to work on the more complicated
side first.
2. Look for trigonometric substitutions involving the
basic identities that may help simplify things.
3. Look for algebraic operations, such as adding fractions,
the distributive property, or factoring, that may simplify
the side you are working with or that will at least lead to
an expression that will be easier to simplify.
4. If you cannot think of anything else to do, change
everything to sines and cosines and see if that helps.
5. Always keep an eye on the side you are not working
with to be sure you are working toward it. There is a
certain sense of direction that accompanies a
successful proof.
6. Practice, practice, practice!
Prove
𝒄𝒐𝒕𝑨(𝟏 + 𝒕𝒂𝒏𝟐 𝑨)
= 𝒄𝒔𝒄𝟐 𝑨
𝒕𝒂𝒏𝑨
𝐜𝐨𝐭𝐀(𝐬𝐞𝐜 𝟐 𝐀)
𝐭𝐚𝐧𝐀
= 𝒄𝒔𝒄𝟐 𝑨
Pythagorean Relationship
𝑐𝑜𝑠𝐴
1
( 2 )
𝑠𝑖𝑛𝐴 𝑐𝑜𝑠 𝐴
𝑠𝑖𝑛𝐴
𝑐𝑜𝑠𝐴
Definition of trig Functions
2
=𝑐𝑠𝑐 𝐴
1
𝑠𝑖𝑛𝐴𝑐𝑜𝑠𝐴 = 𝑐𝑠𝑐 2 𝐴
𝑠𝑖𝑛𝐴
𝑐𝑜𝑠𝐴
Reduce
𝑐𝑜𝑠𝐴
2
=𝑐𝑠𝑐
𝐴
2
𝑠𝑖𝑛 𝐴𝑐𝑜𝑠𝐴
1
2
=
𝑐𝑠𝑐
𝐴
𝑠𝑖𝑛2 𝐴
2
2
𝑐𝑠𝑐 𝐴 = 𝑐𝑠𝑐 𝐴
Reduce
Def of trig function.
Practice Problems Day 2
Sec 8- Written Exercises page 321
#13-19 odds; 29-35 odds
Exit Question: #3b the handout.
A complete, step by step solution must be
included.
Using the identities you now know,
find the trig value.
1.) If cosθ = 3/4, find secθ
2.) If cosθ = 3/5, find cscθ.
sin 2   cos 2   1
1
1
4
sec  


cos 3
3
4
2


3
sin 2      1
 5
25 9
sin 2  

25 25
16
2
sin  
25
4
sin   
5
csc  
1
1
5


sin   4
4
5
3.) sinθ = -1/3, find tanθ
tan 2  1  sec 2 
tan 2  1  (3) 2
tan   2 2
tan 2   8
tan 2   8

4.) secθ = -7/5, find sinθ

Simplifing Trigonometric Expressions
c)
(1 + tan
x)2
- 2 sin x sec x
1
cos x
sin x
2
 1  2 tan x  tan x  2
cos x
 (1  tan x)  2 sin x
2
 1  tan2 x  2tanx  2 tanx
 sec2 x
d)
csc x
tan x  cot x
1

sin x
sin x cos x

cos x sin x
1

sin x
sin 2 x  cos 2 x
sin xcos x
1

sin x
1
sin x cos x
1
sin x cos x


sin x
1
 cos x
Simplify each expression.
1
 cos x 
cos x
  sin x
 sin x 
sin 
cos
sin 
 1  sin x 
 cos x


 sin x  cos x 
1
sin 

sin  cos 
1

1
 sec 
cos
cos 2 x sin 2 x

sin x
sin x
cos 2 x  sin 2 x
sin x
1
 csc x
sin x
Simplifying trig Identity
Example1: simplify
tanxcosx
sin
x
tanx cosx
cos x
tanxcosx = sin x
Simplifying trig Identity
Example2: simplify
sec x
csc x
1
cos
sec x
csc
1x
sin x
=
1
sinx
x
cos x
1
=
sin x
cos x
= tan x
Simplifying trig Identity
Example2: simplify
cos2x - sin2x
cos x
cos2x - sin
1 2x
cos x
= sec x
Example
Simplify:
= cot x (csc2 x - 1)
Factor out cot x
= cot x (cot2 x)
Use pythagorean identi
= cot3 x
Simplify
Example
Simplify:
= sin x (sin x) + cos x
cos x
cos
x
2
= sin x + (cos x)cos x
cos x
= sin2 x + cos2x
cos x
=
1
cos x
= sec x
Use quotient identity
Simplify fraction with
LCD
Simplify numerator
Use pythagorean iden
Use reciprocal identity
Your Turn!
Combine
fraction
Simplify the
numerator
Use
pythagorean
identity
Use Reciprocal
Identity
Practice
1
One way to use identities is to simplify expressions involving trigonometric
functions. Often a good strategy for doing this is to write all trig functions in
terms of sines and cosines and then simplify. Let’s see an example of this:
substitute using each
identity
sin x
tan x 
cos x
tan x csc x
Simplify:
sec x
simplify
sin x 1

 cos x sin x
1
cos x
1
 cos x
1
cos x
1
1
csc x 
sin x
1
sec x 
cos x
Another way to use identities is to write one function in terms of another
function. Let’s see an example of this:
Write the following expression
in terms of only one trig function:
cos x  sin x  1
2
= 1  sin 2 x  sin x  1
This expression involves both sine and
cosine. The Fundamental Identity makes a
connection between sine and cosine so we
can use that and solve for cosine squared
and substitute.
=  sin 2 x  sin x  2
sin 2 x  cos 2 x  1
cos 2 x  1  sin 2 x
(E) Examples
• Prove tan(x) cos(x) = sin(x)
LS  tan x cos x
sin x
LS 
cos x
cos x
LS  sin x
 LS  RS
38
(E) Examples
• Prove tan2(x) = sin2(x) cos-2(x)
RS  sin
2
x cos 2 x
1

2
RS  sin x  

2
cos
x


1
2
RS  sin x 
cos x 2
RS 
sin x 2
cos x 2
 sin x 
RS  

 cos x 
RS  tan 2 x
 RS  LS
2
39
(E) Examples
• Prove
tan x 
1
1

tan x sin x cos x
1
tan x
sin x
1

sin x
cos x
cos x
sin x
cos x

cos x
sin x
sin x sin x  cos x cos x
cos x sin x
2
sin x  cos2 x
cos x sin x
1
cos x sin x
 RS
LS  tan x 
LS 
LS 
LS 
LS 
LS 
 LS
40
(E) Examples
• Prove
LS
LS
LS
LS
sin 2 x
 1  cos x
1  cos x
sin 2 x

1  cos x
1  cos2 x

1  cos x
(1  cos x )(1  cos x )

(1  cos x )
 1  cos x
 LS  RS
41