Download Ch 7 – Trigonometric Identities and Equations

Basic Trigonometric
Identities and
Equations
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
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
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

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 
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
20
(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
21
(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
22
(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
23